Clustered Graph Coloring and Layered Treewidth
aa r X i v : . [ m a t h . C O ] F e b Clustered coloring of graphs excluding a subgraph ∗ Chun-Hung Liu † Department of MathematicsTexas A&M UniversityTexas, USA [email protected]
David R. Wood ‡ School of MathematicsMonash UniversityMelbourne, Australia [email protected]
February 20, 2020 ∗ This material is based upon work supported by the National Science Foundation under Grant No. DMS-1664593 and DMS-1929851. † Partially supported by NSF under Grant No. DMS-1664593 and DMS-1929851. ‡ Research supported by the Australian Research Council. bstract A graph coloring has bounded clustering if each monochromatic component has bounded size.This paper studies clustered coloring, where the number of colors depends on an excluded subgraph.This is a much weaker assumption than previous works, where typically the number of colors dependson an excluded minor. The excluded subgraph is typically a complete bipartite graph K s,t with s t . The case s = 1 is already interesting, since a graph has no K ,t -subgraph if and only ifit has maximum degree less than t . Many of our theorems generalize known results for graphs ofbounded maximum degree to the setting of an excluded K s,t -subgraph. We consider graph classeswith bounded treewidth, excluding a minor, excluding an odd minor, excluding a subdivision, orbounded layered treewidth. Bounded treewidth:
For graphs of bounded treewidth (or equivalently, excluding a planarminor) and with no K s,t -subgraph, we prove ( s + 1) -choosability with bounded clustering, which isbest possible (even for the weaker notion of colorability). Excluded minor:
For graphs excluding a fixed minor and with no K s,t -subgraph, we prove ( s + 2) -colorability with bounded clustering. The number of colors here is best possible. In addition,we show that graphs with no K s +1 -minor are ( s + 1) -colorable with bounded clustering, which iswithin one color of the clustered analogue of Hadwiger’s conjecture. Excluded odd minor:
For graphs excluding a fixed odd minor and with no K s,t -subgraph, weprove (2 s + 1) -colorability with bounded clustering, generalizing a result of the first author and Oumwho proved the case s = 1 . Moreover, at least s − color classes are stable sets. We consider theclustered version of the Gerards–Seymour conjecture, and prove that graphs with no odd K s +1 -minorare (8 s − -colorable with bounded clustering, which improves on previous such bounds. Excluded subdivision:
We prove that graphs with no K s +1 -subdivision are max { s − , } -colorable with bounded clustering. This is the first O ( s ) bound on the clustered chromatic numberof such graphs. This theorem is in marked contrast to the result of Erdős and Fajtlowicz showingthat the maximum chromatic number of such graphs is Ω( s / log s ) (disproving Hajós’ conjecture).We prove several other results for graphs excluding an almost ( -subdivision of a graph H , whichis a subdivision of H where at most one edge is subdivided more than once. We prove that graphsexcluding a fixed minor and containing no almost ( -subdivision of K s +1 are ( s +1) -colorable withbounded clustering. And we show that graphs with bounded treewidth and containing no almost ( -subdivision of K s +1 are s -choosable with bounded clustering. The number of colors here isoptimal. This implies the clustered analogue of Hajós’ conjecture for graphs of bounded treewidth. Bounded layered treewidth:
Layered treewidth is a key tool in the above proofs, and isof independent interest since many natural graph classes have bounded layered treewidth. Thisincludes planar graphs, graphs of bounded Euler genus, apex minor-free graphs, graphs embeddableon a fixed surface with a bounded number of crossings per edge. Our main theorem here says thatgraphs with bounded layered treewidth and with no K s,t -subgraph are ( s +2) -colorable with boundedclustering. In the s = 1 case, we obtain polynomial bounds on the clustering. This greatly improvesa corresponding result of Esperet and Joret for graphs of bounded genus and bounded maximumdegree. The s = 3 case implies that every graph that has a drawing on a fixed surface with a boundednumber of crossings per edge is 5-colorable with bounded clustering.We emphasize that the paper is not merely a collection of disparate results. We in fact presenta series of theorems, where each proof depends on previous theorems and techniques in the series.In particular, the results for an excluded minor or an excluded odd minor depend on the results fortreewidth and layered treewidth, and the results for an excluded subdivision depend on the resultsfor an excluded minor. ontents K s,t -Minor-Free Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 H -Minor-Free Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Colin de Verdiére Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Bounded Treewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Excluded Odd Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 Layered Treewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.9 Layered Treewidth Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.10 Organization of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 K s,t -Subgraph . . . . . . . . . . . . . . . . . . . . 21 ( -Subdivisions . . . . . . . . . . . . . . . . . . . . . . . 139 Introduction
This paper considers graph colorings where the condition that adjacent vertices are as-signed distinct colors is relaxed. Instead, we require that every monochromatic component hasbounded size (for a given graph class). Define a coloring of a graph G to simply be a functionthat assigns one color to each vertex of G . Consider a coloring c of a graph G . A subgraph H of G is c -monochromatic if all the vertices in H are assigned the same color. When c isclear, we simply say H is monochromatic . A c -monochromatic component of G is a connectedcomponent of a subgraph of G induced by all the vertices assigned the same color by c . When c is clear, we simply write monochromatic component . A coloring has clustering λ if everymonochromatic component has at most λ vertices. Note that a coloring with clustering λ isequivalent to a partition of the graph into induced subgraphs with components of size at most λ . Our focus is on minimizing the number of colors, with small monochromatic componentsas a secondary goal. The clustered chromatic number of a graph class G is the minimum k ∈ N such that for some integer λ , every graph in G is k -colorable with clustering λ . Therehave been several recent papers on this topic [10, 18, 22, 23, 33, 34, 37, 40, 45, 46, 49, 52];see [78] for a survey.What distinguishes the present paper from previous contributions is that the number ofcolors is determined by an excluded subgraph . Excluding a subgraph is a much weakerassumption than excluding a minor, which is a typical assumption in previous results. Inparticular, we consider graphs with no complete bipartite K s,t -subgraph plus various otherstructural properties, and prove that every such graph is colorable with bounded clustering,where the number of colors only depends on s . All the dependence on t and the structuralproperty in question is hidden in the clustering function. The structural properties we considerinclude bounded treewidth, bounded layered treewidth, excluding a fixed minor, excluding afixed odd minor and excluding a fixed subdivision.Note that the case s = 1 is already interesting, since a graph has no K ,t -subgraph if andonly if it has maximum degree less than t . Many of our theorems generalize known resultsfor graphs of bounded maximum degree to the setting of an excluded K s,t -subgraph.Also note that for s > , no result with bounded clustering is possible for graphs with no K s,t -subgraph, without making some extra assumption. In particular, for every graph H thatcontains a cycle, and for all k, λ ∈ N , if G is a graph with chromatic number greater than kλ and girth greater than | V ( H ) | (which exists [21]), then G contains no H -subgraph and G isnot k -colorable with clustering λ , for otherwise G would be kλ -colorable. While this paper is the first to study clustered coloring of graphs excluding a subgraph, improper col-orings of graphs excluding a subgraph and with some assumed structural property were previously studiedin the context of defective coloring by Ossona de Mendez, Oum and the second author [55]. That is, eachmonochromatic component has bounded maximum degree, which is a considerably weaker property than hav-ing bounded clustering. Ossona de Mendez et al. [55] considered graphs with no K ∗ s,t -subgraph, where K ∗ s,t is the graph obtained from K ∗ s,t by adding a 1-subdivision of K s on the part of the bipartition of K s,t with s vertices. Consider a graph G containing no K ∗ s,t -subgraph and with the property that if the 1-subdivisionof any graph H is a subgraph of G , then H has bounded average degree. Their main results says that G is s -choosable with bounded defect. Our results are for graphs containing no K s,t subgraph where s t andthe dependence on t is not significant. Since K s,t + ( s ) contains K ∗ s,t as a subgraph, the graphs containingno K ∗ s,t subgraph are a subset of the graphs containing no K s,t + ( s ) subgraph. Hence, excluding K ∗ s,t is notsignificantly more general than excluding K s,t .
4n the following discussion, missing definitions can be found in Section 2.
We start by considering the clustered analogue of Hadwiger’s Conjecture. A graph H isa minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G bycontracting edges. Hadwiger’s conjecture [30] asserts that every graph with no K s +1 -minor hasa proper s -coloring (that is, distinct adjacent vertices are assigned distinct colors). For s the conjecture is easy. Hadwiger [30] and Dirac [14] independently proved the s = 3 case.Wagner [76] proved that Hadwiger’s conjecture with s = 4 is equivalent to the Four ColorTheorem [2, 60]. And Robertson, Seymour, and Thomas [66] proved Hadwiger’s conjecture for s = 5 . The conjecture remains open for s > . Hadwiger’s conjecture is widely considered tobe one of the most important open problems in graph theory. For many years, the best upperbound on the chromatic number of K s +1 -minor-free graphs was O ( s √ log s ) independentlydue to Kostochka [42, 43] and Thomason [73, 74]. Recently, Norin and Song [53] improvedthis bound to O ( s (log s ) . ) , which was further improved by Postle [57] to O ( s (log s ) β ) forany β > . It is open whether every graph with no K s +1 -minor is O ( s ) -colorable. See thesurvey by Seymour [70] for more on Hadwiger’s conjecture.One way to approach Hadwiger’s conjecture is to allow clustered colorings. Kawarabayashiand Mohar [40] first proved a O ( s ) upper bound on the clustered chromatic number of K s +1 -minor-free graphs. The number of colors has since been steadily improved, as shown inTable 1, where λ ( s ) is some large unspecified function.Table 1: Clustered coloring of K s +1 -minor-free graphsnumber of colors clustering choosabilityKawarabayashi and Mohar [40] ⌈ ( s + 1) ⌉ λ ( s ) yesWood [77] ⌈ s +42 ⌉ λ ( s ) yesEdwards, Kang, Kim, Oum, and Seymour [19] s λ ( s ) Liu and Oum [46] s λ ( s ) Norin [51] s λ ( s ) Van den Heuvel and Wood [34] s ⌈ s − ⌉ Dvořák and Norin [18] s λ ( s ) Theorem 1.1 s + 2 λ ( s ) Corollary 1.5 s + 1 λ ( s ) It remains open whether graphs with no K s +1 -minor are s -colorable with bounded clus-tering . Note that s colors would be best possible for any fixed clustering value. That is, forall s > and λ there is a graph G with no K s +1 -minor such that every ( s − -coloring of G This result depended on a result announced by Norine and Thomas [54, 72] which has not yet beenwritten. See [70] for some of the details. Dvořák and Norin [18] have announced that a forthcoming paper proves that graphs with no K s +1 -minor are s -colorable (in fact, s -choosable) with bounded clustering. This would be the clustered analogue ofHadwiger’s Conjecture. Their result is incomparable with our general result in Theorem 1.2. λ vertices .The previously best known upper bound on the clustered chromatic number of K s +1 -minor-free graphs is s [18, 34, 51]. We prove the following bound, which is within two colorsof best possible. Theorem 1.1.
For every s ∈ N , there exists λ ∈ N such that every graph with no K s +1 -minoris ( s + 2) -colorable with clustering λ . We in fact prove the following stronger result where the number of colors only dependson an excluded K s,t -subgraph (see Corollary 1.5 for another strengthening of Theorem 1.1).Indeed, the number of colors only depends on s . The dependence on t and the excluded minoris hidden in the clustering function. Theorem 1.2.
For all s, t ∈ N and for every graph H , there exists λ ∈ N such that everygraph with no H -minor and with no K s,t -subgraph is ( s + 2) -colorable with clustering λ . Theorem 1.2 is proved in Section 5.3.Since every graph with no K s +1 -minor has no K s,s -subgraph, Theorem 1.1 is an immediatecorollary of Theorem 1.2. While Theorem 1.1 is of substantial interest, we emphasize thatour main results are for graph classes excluding a K s,t -subgraph. As mentioned earlier, onemotivation for this line of research is that a graph contains no K ,t -subgraph if and only if ithas maximum degree less than t . So Theorem 1.2 generalizes a result by the first author andOum [46] who proved the s = 1 case, answering a question of Esperet and Joret [22].While Theorem 1.1 is within two colors of the conjectured answer, we now show thatthe number of colors in Theorem 1.2 is best possible. The proof is a variation on the wellknown “standard” example; see [78]. We claim that for all s, λ ∈ N there is a graph G s withno K s +4 -minor and with no K s,s +6 -subgraph, such that every ( s + 1) -coloring of G s has amonochromatic component on at least λ vertices. We proceed by induction on s . In thebase case, s = 1 , let G be the λ × λ triangular grid graph. Then G has no K -minorsince it is planar, and G has no K , -subgraph since it has maximum degree 6. By the HexLemma [28], every 2-coloring of G has a monochromatic path on λ vertices, as claimed. Nowassume the claim for G s − . Let G s be obtained from λ disjoint copies of G s − by adding anew vertex v adjacent to all other vertices. Each component of G s − v is a copy of G s − . If G s contains a K s +4 -minor, then some component of G s − v contains a K s +3 -minor, which is acontradiction. Thus G s contains no K s +4 -minor. Similarly, if G s contains a K s,s +6 -subgraph,then G s − v contains a K s − ,s +6 or K s,s +5 -subgraph, both of which contain K s − , ( s − , whichis a contradiction. Thus G s contains no K s,s +6 -subgraph. Now consider an ( s + 1) -coloringof G s . Say v is blue. If every component of G s − v has a blue vertex, then the blue subgraphcontains a star on λ + 1 vertices, and we are done. Otherwise, some component X of G s − v has no blue vertex, and thus has only s colors. By induction, X and hence G s contains amonochromatic component with at least λ vertices, as desired. Edwards et al. [19] proved the following stronger lower bound: for all s > and c there is a graph G withno K s +1 -minor such that every ( s − -coloring of G has a monochromatic component with maximum degreegreater than c . Conversely, Edwards et al. [19] proved that every graph with no K s +1 -minor is s -colorablesuch that each monochromatic component has maximum degree O ( s log s ) . This degree bound was improvedto O ( s ) by van den Heuvel and the second author [34]. .2 Hajós’ Conjecture A subdivision of a graph H is any graph obtained from H by repeatedly applying thefollowing operation: delete an edge vw , and add a new vertex only adjacent to v and w . Ineffect, the edges of G are replaced by internally disjoint paths between their ends. A graph G contains a subdivision of H if some subgraph of G is isomorphic to a subdivision of H .In the 1940s, Hajós conjectured that every graph containing no subdivision of the completegraph K s +1 is s -colorable; see [48, 70, 75]. Dirac [14] proved the conjecture for s .It is open for s ∈ { , } , which would imply the Four Color Theorem. Catlin [6] presentedcounterexamples for all s > , and Erdős and Fajtlowicz [20] proved that the conjecture is falsefor almost all graphs. Indeed, they showed that there exist graphs with no K s +1 -subdivisionand with chromatic number Ω( s / log s ) . The best upper bound on the number of colors is O ( s ) , independently due to Bollobás and Thomason [5] and Komlós and Szemerédi [41]; see[25] for a related result. See [48, 75] for more explicit counterexamples and further discussionof connections to other areas of graph theory.We prove several positive results in the direction of weakenings of Hajós’ conjecture. Mostof these results actually hold (in some sense) for more general classes of graphs than those withno K s +1 -subdivision, as we now explain. The -subdivision of a graph H is the subdivisionof H where every edge is subdivided exactly once. A ( -subdivision of H is a subdivisionof H where every edge is subdivided at most once. An almost ( -subdivision of H isa subdivision of H , where at most one edge is subdivided more than once. Most of ourresults say that all graphs containing no almost ( -subdivision of K s +1 , plus some otherproperties, are s + O (1) -colorable with bounded clustering. Note that every graph containingno almost ( -subdivision of K s +1 contains no subdivision of K s +1 .The following is our first result for excluded subdivisions. It provides a tight Hajós-typeresult for clustered coloring of graphs with bounded treewidth. Treewidth is a graph invariantthat measures how similar a given graph is to a tree; it is a key parameter in algorithmic andstructural graph theory; see Section 2 for the definition and see [4, 31, 58, 59] for surveys. Theorem 1.3.
For all s, w ∈ N , there exists λ ∈ N such that every graph with treewidth atmost w and containing no almost ( -subdivision of K s +1 is s -choosable with clustering λ . The notion of s -choosable with bounded clustering is defined in Section 2. Note that everygraph that is s -choosable with bounded clustering is also s -colorable with bounded clustering.This shows that the number of colors in Theorem 1.3 is best possible in the following strongsense: for all s ∈ N and λ ∈ N there is a graph G with treewidth at most s − (and thuswith no subdivision of K s +1 ), such that every ( s − -coloring of G has a monochromaticcomponent with at least λ vertices; see [78]. In particular, at least s colors are required evenfor this weakening of Hajós’ conjecture.The assumption of bounded treewidth in Theorem 1.3 is equivalent to saying that thegraph excludes a planar graph as a minor by Robertson and Seymour’s Grid Minor The-orem [62]. What if we exclude a general graph as a minor? Our next result answers thisquestion (with one more color). Interestingly, the number of colors does not depend on theexcluded minor. Theorem 1.4.
For every s ∈ N and every graph H , there exists λ ∈ N such that every graphcontaining no H -minor and containing no almost ( -subdivision of K s +1 is ( s +1) -colorablewith clustering λ . H = K s +1 ) has the following corollary for graphs excluding a minor,which is within one color of a clustered analogue of Hadwiger’s Conjecture. Corollary 1.5.
For every s ∈ N there exists λ ∈ N such that every graph containing no K s +1 -minor is ( s + 1) -colorable with clustering λ . Corollary 1.5 improves the bound of s + 2 in Theorem 1.1 (implied by Theorem 1.2) to s + 1 . It should be noted that the proof of Theorem 1.2 is an essential ingredient for the proofof Theorem 1.4 and Corollary 1.5.Our next result relaxes the assumption that the graph contains no H -minor, and insteadassumes that it contains no H -subdivision. The price paid is an increase in the number ofcolors, depending only on the maximum degree of H . Theorem 1.6.
For every s ∈ N and every graph H with maximum degree d ∈ N , there exists λ ∈ N such that every graph with no H -subdivision and no almost ( -subdivision of K s +1 is max { s + 3 d − , } -colorable with clustering λ . The next theorem relaxes the assumption of no almost ( -subdivision of K s +1 , andinstead assumes the graph contains no K s,t -subgraph. Interestingly the number of colors doesnot depend on t . Note that K s,t contains a K s +1 -subdivision where every edge is subdividedat most once, when t is sufficiently large. Theorem 1.7.
For s, t, d ∈ N and every graph H of maximum degree d , there exists λ ∈ N such that every graph with no K s,t -subgraph and no H -subdivision is max { s + 3 d − , } -colorable with clustering λ . We remark that all of the above theorems forbid ( -subdivisions of K s +1 or subdi-visions of H . That is, we forbid a subdivision of a graph where some edge is allowed tobe subdivided arbitrarily many times. This condition is required since there are graphs ofarbitrarily high girth and arbitrarily high chromatic number [21], which therefore require ar-bitrarily many colors for any fixed clustering value; this shows that excluding finitely manygraphs as subgraphs cannot ensure any upper bound on the number of colors.Our final theorem simply excludes a K s +1 -subdivision. This is the first O ( s ) bound onthe clustered chromatic number of the class of graphs excluding a K s +1 -subdivision. Theorem 1.8.
For every s ∈ N , there exists λ ∈ N such that every graph containing no K s +1 -subdivision is max { s − , } -colorable with clustering λ . We now compare the above theorems with Hajós’ conjecture. First note that Theo-rems 1.3–1.6 are stronger than Hajós’ conjecture in the sense that they only exclude an al-most ( -subdivision of K s +1 , whereas Hajós’ conjecture excludes all subdivisions of K s +1 .Moreover, Theorem 1.3 also holds in the stronger setting of choosability. On the other hand,Theorems 1.3–1.8 are weaker than Hajós’ conjecture in the sense that they have boundedclustering rather than a proper coloring. However, such a weakening is unavoidable sinceHajós’ conjecture is false! Indeed, the proof of the theorem of Erdős and Fajtlowicz [20] men-tioned above shows that, for a suitable constant c , almost every graph on cs vertices containsno subdivision of K s +1 and has chromatic number Ω( s / log s ) . Trivially, such a graph hastreewidth at most cs and contains no K cs -minor. Thus the clustering function in all of theabove theorems is at least Ω( s/ log s ) . 8 .3 K s,t -Minor-Free Graphs Consider graphs with no K s,t -minor for s t . Van den Heuvel and the second author [34]observed that results of Edwards et al. [19] and Ossona de Mendez et al. [55] imply that suchgraphs are s -colorable with bounded clustering, which was improved to s + 2 by Dvořákand Norin [18]. Theorem 1.2 immediately implies the following further improvement: Corollary 1.9.
For all s, t ∈ N , there exists λ ∈ N such that every graph with no K s,t -minoris ( s + 2) -colorable with clustering λ . The best known lower bound on the clustered chromatic number of K s,t -minor-free graphsis s + 1 , due to van den Heuvel and the second author [34]. It is open whether every K s,t -minor-free graph is ( s + 1) -colorable with bounded clustering. Van den Heuvel and the secondauthor [34] proved this in the s = 2 case. H -Minor-Free Graphs Now consider the clustered chromatic number of H -minor-free graphs, for an arbitrarygraph H . A vertex-cover of a graph H is a set S ⊆ V ( H ) such that H − S has no edges.Suppose that H has a vertex-cover of size s . Then H is a minor of K s, | V ( H ) |− (obtained bycontracting a matching of size s − in K s, | V ( H ) |− ). So every graph containing no H -minorcontains no K s, | V ( H ) |− -minor. Corollary 1.9 thus implies: Corollary 1.10.
For every s ∈ N and for every graph H that has a vertex-cover of size atmost s , there exists λ ∈ N such that every graph with no H -minor is ( s + 2) -colorable withclustering λ . We now relate this result to a conjecture of Norin, Scott, Seymour and the second au-thor [52] about the clustered chromatic number of H -minor-free graphs. Let T be a rootedtree. The depth of T is the maximum number of vertices on a root–to–leaf path in T . The closure of T is obtained from T by adding an edge between every ancestor and descendentin T . The connected tree-depth of a graph H , denoted by td ( H ) , is the minimum depth of arooted tree T such that H is a subgraph of the closure of T . Norin et al. [52] observed thatfor every graph H and λ ∈ N there is an H -minor-free graph that is not ( td ( H ) − -colorablewith clustering λ ; thus the clustered chromatic number of H -minor-free graphs is at leasttd ( H ) − . On the other hand, Norin et al. [52] conjectured that the class of H -minor-freegraphs has clustered chromatic number at most td ( H ) − , which would be tight for cer-tain graphs H . As evidence for this conjecture, Norin et al. [52] proved that the clusteredchromatic number of H -minor-free graphs is at most td ( H )+1 − .For h ∈ N with h > , a broom of height h is a rooted tree that can be obtained from astar rooted at a leaf by subdividing the edge incident with the root h − times. The closureof a broom of height h has a vertex-cover of size at most h − . Thus Corollary 1.10 can berestated as follows: Corollary 1.11.
For every integer h > , if H is a subgraph of the closure of the broom ofheight h , then there exists λ ∈ N such that every graph with no H -minor is ( h + 1) -colorablewith clustering λ . h equals h . So Corollary 1.11 answers theaforementioned conjecture of Norin et al. [52] (since h + 1 h − ) when the underlying tree T is a broom. The Colin de Verdière parameter µ ( G ) is an important minor-closed graph invariant in-troduced by Colin de Verdière [11, 12]; see [35, 69] for surveys. It is known that µ ( G ) ifand only if G is a union of disjoint paths, µ ( G ) if and only if G is outerplanar, µ ( G ) if and only if G is planar, and µ ( G ) if and only if G is linklessly embeddable. A famousconjecture of Colin de Verdière [11] asserts that every graph G with µ ( G ) s is properly ( s + 1) -colorable, which implies the Four Color Theorem. Since every graph G with µ ( G ) s contains no K s +2 -minor, this conjecture is implied by Hadwiger’s conjecture. However, it iseven open whether every graph G with µ ( G ) s is ( s + 1) -colorable with bounded clustering.Corollary 1.5 implies: Corollary 1.12.
For every s ∈ N there exists λ ∈ N , such that every graph G with µ ( G ) s is ( s + 2) -colorable with clustering λ . We may prove this result by also considering excluded subgraphs. Van der Holst, Lovászand Schrijver [35] proved that µ ( K s,t ) = s + 1 for t > max { s, } . Thus if µ ( G ) s then G contains no K s,t -subgraph (since µ is monotone under taking subgraphs). Theorem 1.2 thenimplies Corollary 1.12. This proof highlights the utility of excluding a subgraph within aminor-closed class. When the excluded minor H is planar (or equivalently, when the graph has boundedtreewidth), Theorem 1.2 is improved as follows. Theorem 1.13.
For all s, t, w ∈ N there exists λ ∈ N such that every graph with treewidthat most w and with no K s,t -subgraph is ( s + 1) -colorable with clustering λ . Note that the number of colors in Theorem 1.13 is best possible: for all s, c ∈ N there isa graph G with treewidth s , with no K s,s +2 -subgraph, and such that every s -coloring of G has a monochromatic component on at least c vertices. The construction is analogous to theconstruction above except that in the base case ( s = 1 ) we use a long path instead of thetriangular grid. This is called a “standard” example in [78].We actually prove the following list-coloring result, which immediately implies Theo-rem 1.13. Theorem 1.14.
For all s, t, w ∈ N , there exists λ ∈ N such that every graph with treewidthat most w and with no K s,t -subgraph is ( s + 1) -choosable with clustering λ . Theorem 1.14 is proved in Sections 3 and 5.1.The case s = 1 of Theorem 1.14 is an unpublished result of the first author (see [70,Theorem 6.4]), which generalizes a result of Alon, Ding, Oporowski, and Vertigan [1] whoproved Theorem 1.13 in the case s = 1 (with much better bounds on λ ).10heorem 1.14 immediately implies results for graphs with bounded treewidth and withno K s,t -minor, although we emphasise that Theorem 1.14 holds in the stronger setting ofan excluded K s,t -subgraph. Since K s, ( s ) +1 contains a ( -subdivision of K s +1 , this in turnimplies that graphs with bounded treewidth and with no ( -subdivision of K s +1 are ( s +1) -choosable with bounded clustering. Theorem 1.3 pushes this result further by showing thatgraphs with bounded treewidth and with no K s +1 -subdivision are s -choosable with boundedclustering. This not only says that a clustered version of Hajós’ conjecture holds for boundedtreewidth graphs, and even holds for choosability, but also implies a result of Dvořák andNorin [18] who proved the same result for graphs with bounded treewidth and with no K s +1 -minor. Gerards and Seymour (see [36, §6.5]) conjectured that every graph with no odd K s +1 -minor is properly s -colorable, which implies Hadwiger’s conjecture. The best known upperbound on the chromatic number of graphs with no odd K s +1 -minor is O ( s √ log s ) , due toGeelen, Gerards, Reed, Seymour, and Vetta [29]. It is open whether such graphs are properly O ( s ) colorable. The first O ( s ) bound on the clustered chromatic number was established byKawarabayashi [39], who proved that every graph with no odd K s -minor is s -colorablewith bounded clustering. The number of colors was improved to s − by Kang andOum [37]. We make the following modest improvement. Theorem 1.15.
For every s ∈ N there exists λ ∈ N such that every graph with no odd K s +1 -minor is (8 s − -colorable with clustering λ . Theorem 1.15 is proved in Section 5.4.More interestingly, we prove the following analogue of Theorem 1.2 for excluded oddminors and excluded subgraphs.
Theorem 1.16.
For all s, t ∈ N and for every graph H there exists λ ∈ N such that everygraph with no odd H -minor and with no K s,t -subgraph is (2 s + 1) -colorable with clustering λ .Moreover, at least s − color classes are stable sets. Theorem 1.16 is proved in Section 5.3.The case of s = 1 in Theorem 1.16 was proved by the first author and Oum [46]. Here,three colors is best possible.Note that no clustered choosability result is possible for graphs excluding an odd minor,since the complete bipartite graph K n,n contains no odd K -minor, but it follows from thework of Kang [38] that for all k, λ ∈ N there exists n ∈ N such that K n,n is not k -choosablewith clustering λ . Layered treewidth is a graph parameter independently introduced by Dujmović, Morin,and Wood [17] and Shahrokhi [71]. It is of interest since several natural graph classes havebounded layered treewidth. For example, Dujmović et al. [17] proved that every planargraph has layered treewidth at most 3; more generally, that every graph with Euler genus11t most g has layered treewidth at most g + 3 ; and most generally, that a minor-closedclass has bounded layered treewidth if and only if it excludes some apex graph as a minor.Layered treewidth is of interest beyond minor-closed classes, since as described below, thereare several natural graph classes that have bounded layered treewidth but contain arbitrarilylarge complete graph minors.It is easily seen that the clustered chromatic number of the class of graphs with layeredtreewidth w equals w , and indeed, every such graph is properly w -colorable (see Section 2for the proof). On the other hand, we prove that for clustered coloring of graphs excluding a K s,t -subgraph in addition to having bounded layered treewidth, only s + 2 colors are needed(no matter how large the upper bound on layered treewidth). Theorem 1.17.
For all s, t, w ∈ N there exists λ ∈ N such that every graph with layeredtreewidth at most w and with no K s,t -subgraph is ( s + 2) -colorable with clustering λ . Theorem 1.17 is proved in Section 4.6. The case s = 1 in Theorem 1.17 is of particularinterest, since it applies for graphs of maximum degree ∆ < t . It implies that graphs ofbounded Euler genus and of bounded maximum degree are 3-colorable with bounded cluster-ing, which was previously proved by Esperet and Joret [22]. The clustering function provedby Esperet and Joret [22] is roughly O (∆
32∆ 2 g ) , for graphs with Euler genus g and maximumdegree ∆ . While Esperet and Joret [22] made no effort to reduce this function, their methodwill not lead to a sub-exponential clustering bound. In the case s = 1 we give a different andsimpler proof of Theorem 1.17 with the following polynomial clustering bound. Theorem 1.18.
For all w, ∆ ∈ N , every graph with layered treewidth at most w and maximumdegree at most ∆ is 3-colorable with clustering O ( w ∆ ) . In particular, every graph withEuler genus at most g and maximum degree at most ∆ is 3-colorable with clustering O ( g ∆ ) . Note that the proof of Theorem 1.18 (presented in Section 4.1) is relatively simple, avoidingmany technicalities that arise when dealing with graph embeddings. This proof highlightsthe utility of layered treewidth as a general tool.The number of colors in Theorem 1.18 is best possible since the Hex Lemma [28] saysthat every 2-coloring of the n × n planar triangular grid (which is easily seen to have layeredtreewidth 2 and maximum degree 6) contains a monochromatic path of length at least n .Further motivation for Theorem 1.17 is that it is a critical ingredient in the proofs ofour results about clustered colorings of graphs excluding a minor (Theorem 1.2), odd minor(Theorem 1.16), or subdivision (Theorems 1.4 and 1.7). For these proofs we actually needthe following stronger result proved in Section 4.7 (where the ξ = 0 case is Theorem 1.17). Theorem 1.19.
For all s, t, w ∈ N and ξ ∈ N , there exists λ ∈ N such that if G is a graphwith no K s,t -subgraph and G − Z has layered treewidth at most w for some Z ⊆ V ( G ) with | Z | ξ , then G is ( s + 2) -colorable with clustering λ . Note that the number of colors in Theorem 1.19 is best possible for s : Suppose thatthe theorem holds with s = 2 , t = 7 , w = 2 and ξ = 1 , but with only 3 colors. Let n ≫ λ .Let G be obtained from the n × n triangular grid by adding one dominant vertex v . Then G contains no K , -subgraph, and it is easily seen that G − v has layered treewidth 2. Byassumption, G is 3-colorable with clustering λ . Say v is blue. Since v is dominant, at most12 rows or columns contain a blue vertex. The non-blue induced subgraph contains an λ × λ triangular grid (since n ≫ λ ). This contradicts the Hex Lemma mentioned above.We remark that the class of graphs mentioned in Theorem 1.19 is more general than theclass of graphs with bounded layered treewidth. For example, the class of graphs that can bemade planar (and hence bounded layered treewidth) by deleting one vertex contains graphsof arbitrarily large layered treewidth. We now give several examples of graph classes with bounded layered treewidth, for whichTheorems 1.17 and 1.18 give interesting results. ( g, k ) -Planar Graphs A graph is ( g, k ) -planar if it can be drawn in a surface of Euler genus at most g with atmost k crossings on each edge (assuming no three edges cross at a single point). Such graphscan contain arbitrarily large complete graph minors, even in the g = 0 and k = 1 case [15].On the other hand, Dujmović et al. [15] proved that every ( g, k ) -planar graph has layeredtreewidth at most (4 g + 6)( k + 1) . Theorem 1.18 then implies: Corollary 1.20.
For all g, k, ∆ ∈ N , every ( g, k ) -planar graph with maximum degree at most ∆ is 3-colorable with clustering O ( g k ∆ ) . Now consider ( g, k ) -planar graphs without the assumption of bounded maximum degree.The second author [78] proved that such graphs are 12-colorable with clustering bounded bya function of g and k . Ossona de Mendez et al. [55] proved that every ( g, k ) -planar graphcontains no K ,t -subgraph for some t = O ( kg ) . Thus Theorem 1.17 with s = 3 implies thatthis bound of 12 can be reduced to 5: Corollary 1.21.
For all g, k ∈ N , there exists λ ∈ N , such that every ( g, k ) -planar graph is5-colorable with clustering λ . Corollary 1.21 highlights the utility of excluding a K s,t -subgraph. It also generalizes atheorem of Esperet and Ochem [23] who proved the k = 0 case, which says that every graphwith bounded Euler genus is 5-colorable with bounded clustering. Note that Dvořák andNorin [18] recently proved that every graph with bounded Euler genus is 4-colorable (infact, 4-choosable) with bounded clustering. It is open whether every ( g, k ) -planar graph is4-colorable with bounded clustering. Map Graphs
Map graphs are defined as follows. Start with a graph G embedded in a surface of Eulergenus g , with each face labelled a “nation” or a “lake”, where each vertex of G is incidentwith at most d nations. Let G be the graph whose vertices are the nations of G , where twovertices are adjacent in G if the corresponding faces in G share a vertex. Then G is calleda ( g, d ) -map graph . A (0 , d ) -map graph is called a (plane) d -map graph ; such graphs havebeen extensively studied [7–9, 13, 24]. The ( g, -map graphs are precisely the graphs of Eulergenus at most g (see [9, 15]). So ( g, d ) -map graphs provide a natural generalization of graphsembedded in a surface that allows for arbitrarily large cliques even in the g = 0 case (since13f a vertex of G is incident with d nations then G contains K d ). Dujmović et al. [15] provedthat every ( g, d ) -map graph is ( g, ⌊ ( d − ⌋ ) -planar. Thus Corollary 1.20 implies: Corollary 1.22.
For all g, d, ∆ ∈ N , every ( g, d ) -map graph with maximum degree at most ∆ is 3-colorable with clustering O ( g d ∆ ) . Similarly, Corollary 1.21 implies:
Corollary 1.23.
For all g, d ∈ N , there exists λ ∈ N , such that every ( g, d ) -map graph is5-colorable with clustering λ . It is straightforward to prove that, in fact, if a ( g, d ) -map graph contains a K ,t -subgraph,then t d ( g + 1) . (We omit these details.) Dujmović et al. [15] proved that every ( g, d ) -mapgraph has layered treewidth at most (2 g + 3)(2 d + 1) . Thus Theorems 1.17 and 1.18 can beapplied directly with s = 3 , slightly improving the clustering bounds.It is open whether every ( g, d ) -map graph is 4-colorable with clustering bounded by afunction of g and d . String Graphs A string graph is the intersection graph of a set of curves in the plane with no three curvesmeeting at a single point [26, 27, 44, 56, 67, 68]. For k ∈ N , if each curve is in at most k intersections with other curves, then the corresponding string graph is called a k -string graph .A ( g, k ) -string graph is defined analogously for curves on a surface of Euler genus at most g . Dujmović et al. [16] proved that every ( g, k ) -string graph has layered treewidth at most k − g + 3) . By definition, the maximum degree of a ( g, k ) -string graph is at most k (andmight be less than k since two curves might have multiple intersections). Thus Theorem 1.18implies: Corollary 1.24.
For all g, k ∈ N with k > , every ( g, k ) -string graph is -colorable withclustering O ( g k ) . Section 2 contains definitions of standard terminology and notation.Section 3 contains the first proof of our main result concerning treewidth (Theorem 1.14).The result is reproved in a more general setting in Section 5 on excluded minors. We includethe proof in Section 3 since it is the easiest way for the reader to understand our use of listcolorings and the assumption of having no K s,t -subgraph, both of which are key ideas in theremainder.Section 4 proves our results concerning layered treewidth (Theorems 1.17 to 1.19). Thisproof uses an extension of the list coloring argument used in Section 3.Section 5 proves our results for graphs of bounded treewidth, excluding a minor, or ex-cluding an odd minor (Theorems 1.2 and 1.14 to 1.16). This proof uses several techniquesand results from the previous sections on treewidth and layered treewidth, along with theGraph Minor Structure Theorem of Robertson and Seymour [65].Note that our result for graphs of bounded treewidth (Theorem 1.14) is proved in Section 3using separators, while the proof of the same theorem in Section 5 uses tangles, which are a14ual concept to separators. Tangles are needed here since the proof of Theorem 1.2 uses theGraph Minor Structure Theorem (in the large treewidth case) to describe the structure of agraph excluding a fixed minor relative to a tangle.Section 6 proves our results for graphs excluding a subdivision (Theorems 1.4 and 1.6to 1.8). This proof uses the results from the previous section on excluded minors, along witha structure theorem for excluding subdivisions by the first author and Thomas [47].We conclude by mentioning some open problems in Section 7. We use the following notation. Let N := { , , . . . } and N := { , , , . . . } . For m, n ∈ N ,let [ m, n ] := { m, m + 1 , . . . , n } and [ n ] := [1 , n ] . For a set X let (cid:0) X (cid:1) := {{ v, w } : v, w ∈ X, v = w } .Let G be a graph with vertex-set V ( G ) and edge-set E ( G ) (allowing loops and paralleledges). For v ∈ V ( G ) , let N G ( v ) := { w ∈ V ( G ) : vw ∈ E ( G ) } be the neighborhood of v , and let N G [ v ] := N G ( v ) ∪ { v } . For X ⊆ V ( G ) , let N G ( X ) := S v ∈ X ( N G ( v ) − X ) and N G [ X ] := N G ( X ) ∪ X . Denote the subgraph of G induced by X by G [ X ] .A set S of vertices in a graph G is stable (also called independent ) if no non-loop edge of G has both ends in S .For our purposes, a color is an element of Z . A list-assignment of a graph G is a function L with domain containing V ( G ) , such that L ( v ) is a non-empty set of colors for each vertex v ∈ V ( G ) . For a list-assignment L of V ( G ) , an L -coloring of G is a function c with domain V ( G ) such that c ( v ) ∈ L ( v ) for every v ∈ V ( G ) . So an L -coloring has clustering λ if everymonochromatic component has at most λ vertices. A list-assignment L of a graph G is an ℓ -list-assignment if | L ( v ) | > ℓ for every vertex v ∈ V ( G ) . A graph is ℓ -choosable with clustering λ if G is L -colorable with clustering λ for every ℓ -list-assignment L of G .The following is an alternative definition of graph minor. Recall that we allow loops andparallel edges. For a graph H , an H -minor of a graph G is a map α with domain V ( H ) ∪ E ( H ) such that: • For every h ∈ V ( H ) , α ( h ) is a nonempty connected subgraph of G (called a branch set ), • If h and h are different vertices of H , then α ( h ) and α ( h ) are disjoint. • For each edge e = h h of H , α ( e ) is an edge of G with one end in α ( h ) and one endin α ( h ) ; furthermore, if h = h , then α ( e ) ∈ E ( G ) − E ( α ( h )) . • If e , e are distinct edges of H , then α ( e ) = α ( e ) .Then α is an odd H -minor if there exists a 2-coloring c of S h ∈ V ( H ) α ( h ) such that c | α ( h ) is aproper 2-coloring of α ( h ) , and for every edge e of H , the ends of α ( e ) receive the same colorin c . See [29, 37, 39] for work on odd minors.A tree-decomposition of a graph G is a pair ( T, X = ( X x : x ∈ V ( T ))) , where T is atree, and for each node x ∈ V ( T ) , X x is a non-empty subset of V ( G ) called a bag , such thatfor each vertex v ∈ V ( G ) , the set { x ∈ V ( T ) : v ∈ X x } induces a non-empty (connected)subtree of T , and for each edge vw ∈ E ( G ) there is a node x ∈ V ( T ) such that v, w ∈ X x .A path-decomposition is a tree-decomposition whose underlying tree is a path. The width ofa tree-decomposition ( T, X ) is max {| X x | − x ∈ V ( T ) } . The treewidth of a graph G is the15inimum width of a tree-decomposition of G . For each k ∈ N , the graphs with treewidth atmost k form a minor-closed class. Robertson and Seymour [62] proved that a minor-closedclass of graphs has bounded treewidth if and only if some planar graph is not in the class.A separation of a graph G is an ordered pair ( A, B ) of edge-disjoint subgraphs of G with A ∪ B = G . The order of ( A, B ) is | V ( A ∩ B ) | . Every graph with treewidth k has a ‘balanced’separation of order at most k + 1 ; see Lemma 3.2 for a precise statement.A layering of a graph G is an ordered partition ( V , . . . , V n ) of V ( G ) into (possibly empty)sets such that for each edge vw ∈ E ( G ) there exists i ∈ [1 , n − such that { v, w } ⊆ V i ∪ V i +1 . The layered treewidth of a graph G is the minimum nonnegative integer ℓ such that G has a tree-decomposition ( T, X = ( X x : x ∈ V ( T )) and a layering ( V , . . . , V n ) , such that | X x ∩ V i | ℓ for each bag X x and layer V i . This says that the subgraph induced by eachlayer has bounded treewidth, and moreover, a single tree-decomposition of G has boundedtreewidth when restricted to each layer. In fact, these properties hold when considering abounded sequence of consecutive layers.We now determine the clustered chromatic number of the class of graphs with layeredtreewidth w (with no other properties), as mentioned in Section 1.8. Consider a graph G withlayered treewidth w . Say ( V , . . . , V n ) is the corresponding layering. Then G [ S i odd V i ] hastreewidth at most w − , and is thus properly w -colorable. Similarly, G [ S i even V i ] is properly w -colorable. Hence G is properly w -colorable. This bound is best possible since K w haslayered treewidth w . In fact, for all d, w ∈ N , there is a graph G with treewidth w − , suchthat every (2 w − -coloring of G has a vertex of monochromatic degree at least d , implyingthere is a monochromatic component with more than d vertices [78]. Bannister et al. [3]observed that every graph with treewidth k has layered treewidth at most ⌈ k +12 ⌉ (using twolayers), so G has layered treewidth at most w . This says that the clustered chromatic numberof the class of graphs with layered treewidth w equals w , and indeed, every such graph isproperly w -colorable.A surface is a nonnull compact connected -manifold without boundary. Every surface ishomeomorphic to the sphere with k handles (which has Euler genus k ) or the sphere with k cross-caps (which has Euler genus k ). The Euler genus of a graph G is the minimum Eulergenus of a surface in which G embeds; see [50] for more on graph embeddings. This section proves our results for clustered choosability of graphs with bounded treewidthand no K s,t -subgraph (Theorem 1.14). The proofs serve as an introduction to the moreadvanced methods employed later in the paper. Section 3.1 starts with the s = 1 case, whichcorresponds to graphs of bounded maximum degree. Section 3.2 consists of helpful definitionsthat enable the proof to be generalized for higher values of s . Section 3.3 introduces a list-coloring framework that is used throughout the paper. With these tools in hand, we completethe proof of Theorem 1.14 in Section 3.4. This section proves the following previously unpublished result of the first author (see [70,Theorem 6.4]). 16 heorem 3.1.
For all k, ∆ ∈ N there exists λ ∈ N such that every graph with treewidth lessthan k and maximum degree at most ∆ is -choosable with clustering λ . We now sketch the proof of Theorem 3.1. The first idea is to allow for a precolored setof vertices Y of bounded size. First assume that Y is small. Let c , c , . . . , c r be the colorsappearing in Y . First precolor N ( Y ) so that no vertex uses c , then precolor N ( N ( Y )) so thatno vertex uses c , and so on, until the r -th neighborhood of Y is precolored so that no vertexuses c r . Let Y ′ be the set of vertices at distance at most r from Y . Since Y is small and G has bounded maximum degree, | Y ′ | is bounded. Apply induction to obtain a desired coloringof G with Y ′ precolored. Note that in this coloring of G , every monochromatic componentintersecting Y is contained in G [ Y ′ ] so the size is bounded by a function of Y . When Y is largewe use the following separator lemma of Robertson and Seymour [61] to split the probleminto two smaller sub-problems to which we apply induction. Lemma 3.2 ([61, (2.6)]) . For every graph G with treewidth less than k ∈ N and for all S ⊆ V ( G ) , there exists a separation ( G , G ) of G of order at most k , such that if S := S − V ( G ) and S := S − V ( G ) , then | S | | S | and | S | | S | . The following definition is a key to the proof of Theorem 3.1 (and many other results in thepaper). Let η ∈ N and g : N → N be a nondecreasing function. Let L be a list-assignment ofa graph G , and let Y := { v ∈ V ( G ) : | L ( v ) | = 1 } . Then an L -coloring c of G is ( η, g ) -bounded if: • c has clustering at most η g ( η ) , and • the union of the c -monochromatic components intersecting Y contains at most | Y | g ( | Y | ) vertices.The next lemma motivates this definition. Lemma 3.3.
Fix η ∈ N and a nondecreasing function g : N → N . Let G be a graphwith treewidth less than k ∈ N . Let L be a list-assignment for G with Y := { v ∈ V ( G ) : | L ( v ) | = 1 } and | Y | ∈ [12 k, η ] . Let ( G , G ) be a separation of G satisfying Lemma 3.2 with S = Y . For i ∈ { , } , let L i be a list-assignment for G i where L i ( v ) is a subset of L ( v ) for each v ∈ V ( G i ) − V ( G − i ) , and L ( v ) = L ( v ) is a 1-element subset of L ( v ) for each v ∈ V ( G ) ∩ V ( G ) . Assume that G i has an ( η, g ) -bounded L i -coloring for i ∈ { , } . Then G has an ( η, g ) -bounded L -coloring.Proof. Let I := V ( G ) ∩ V ( G ) . Let S := S − V ( G ) and S := S − V ( G ) . By Lemma 3.2, | I | k and | S | | S | and | S | | S | . For i ∈ { , } , let Y i := S i ∪ I ; note that { v ∈ V ( G i ) : | L i ( v ) | = 1 } = Y i and k | Y | = | S | + | S | + | Y ∩ I | | S i | + | Y ∩ I | = 3 | Y i − I | + | Y ∩ I | = 3 | Y i |− | I | + | Y ∩ I | | Y i | . Thus | Y i | > k .By assumption, there is an ( η, g ) -bounded L i -coloring of G i . Let U i be the union ofthe monochromatic components of G i that intersect Y i . Then | V ( U i ) | | Y i | g ( | Y i | ) . Since L ( v ) = L ( v ) is a 1-element subset of L ( v ) , for each v ∈ V ( G ) ∩ V ( G ) , we obtain an L -coloring of G . 17n G , the union of the monochromatic components intersecting Y is a subgraph of U ∪ U , which has size at most | Y | g ( | Y | ) + | Y | g ( | Y | ) . We now prove that | Y | g ( | Y | ) + | Y | g ( | Y | ) | Y | g ( | Y | ) , which implies that the union of the monochromatic componentsintersecting Y has size at most | Y | g ( | Y | ) , as desired.Since | Y | , | Y | > k we have | Y || Y | − k ( | Y | + | Y | ) + 4 k = ( | Y | − k )( | Y | − k ) > k ,implying | Y || Y | > k ( | Y | + | Y | ) and | Y || Y | − k ( | Y | + | Y | ) > . By construction, | Y | + | Y | | Y | + 2 | I | | Y | + 2 k . Thus | Y | > ( | Y | + | Y | − k ) = | Y | + | Y | + 2 | Y || Y | − k ( | Y | + | Y | ) + 4 k > | Y | + | Y | . In particular, | Y | , | Y | | Y | , implying | Y | g ( | Y | ) > | Y | g ( | Y | ) + | Y | g ( | Y | ) , as desired.Finally, we show that the L -coloring of G has clustering η g ( η ) . Let M be a monochromaticcomponent of G . If M intersects only one of G and G , then | V ( M ) | η g ( η ) , as desired.Now assume that M intersects both G and G , implying that M intersects V ( G ∩ G ) .Thus M ∩ G intersects Y and M ∩ G intersects Y . Hence | V ( M ) | | V ( U ) | + | V ( U ) | | Y | g ( | Y | ) + | Y | g ( | Y | ) | Y | g ( | Y | ) , and our L -coloring of G is ( η, g ) -bounded.Theorem 3.1 is then implied by the following lemma. Lemma 3.4.
For all k, ∆ ∈ N there exists η ∈ N and a nondecreasing function g : N → N such that for every graph G of treewidth less than k and with maximum degree at most ∆ ,for every 2-list assignment L of G with |{ v ∈ V ( G ) : | L ( v ) | = 1 }| η , there exists an ( η, g ) -bounded L -coloring of G .Proof. Define η := 24 k ∆ k and define the function g by g ( n ) := 2∆ n for n ∈ N .Suppose for the sake of contradiction that there is a counterexample ( G, L ) . Let Y := { v ∈ V ( G ) : | L ( v ) | = 1 } . Choose a counterexample ( G, L ) such that | V ( G ) | is minimized,and subject to this condition, | Y | is maximized.We first deal with two trivial cases. Case A. Y = ∅ : Let x be a vertex of G . Let L ′ ( x ) be a 1-element subset of L ( x ) . Foreach vertex v ∈ V ( G ) − { x } , let L ′ ( v ) := L ( v ) . Let Y ′ := { x } . Then L ′ is a -list assignmentfor G . By the choice of counterexample, ( G, L ′ ) is not a counterexample. Thus there existsan ( η, g ) -bounded L ′ -coloring of G , which is also an ( η, g ) -bounded L -coloring of G . Hence ( G, L ) is not a counterexample. Now assume that Y = ∅ . Case B. N ( Y ) = ∅ : By the choice of counterexample, ( G − Y, L ) is not a counterexample.Thus G − Y has an ( η, g ) -bounded L -coloring. Assign each vertex v ∈ Y the color in L ( v ) .Since N ( Y ) = ∅ , every monochromatic component intersecting Y is contained in G [ Y ] , whichhas size | Y | | Y | g ( | Y | ) . Thus we have an ( η, g ) -bounded L -coloring of G . Now assumethat N ( Y ) = ∅ .We now deal with the two main cases. Case C. | Y | ∈ [1 , k − : Say Y =: { x , . . . , x | Y | } . For i ∈ [0 , y ] , let Y i be the set ofvertices at distance i from Y . For each i ∈ [1 , | Y | ] and for each vertex v ∈ Y i , let L ′ ( v ) := L ( v ) − L ( x i ) , which has size at least 1. Let L ′ ( v ) := L ( v ) for every vertex at distance at least | Y | + 1 from Y . Let Y ′ := Y ∪ Y ∪ · · · ∪ Y | Y | . Thus | Y ′ | | Y | (1 + ∆ + ∆ + · · · + ∆ | Y | ) y ∆ | Y | η . By the choice of counterexample, ( G, L ′ ) is not a counterexample. Thus G hasan ( η, g ) -bounded L ′ -coloring. This is an L -coloring since L ′ ( v ) ⊆ L ( v ) for every vertex v G . Consider a monochromatic chromatic M of G intersecting Y . Then x i ∈ M for some i ∈ [1 , | Y | ] . By construction, no vertex at distance i from Y is assigned the same color as x i .Thus M ⊆ Y ∪ Y ∪ · · · ∪ Y i − . Hence the union of monochromatic components intersecting Y has size at most | Y ∪ Y ∪ · · · ∪ Y i − | | Y | ∆ i − | Y | ∆ | Y |− | Y | g ( | Y | ) . Case D. | Y | ∈ [12 k, η ] : Then Lemma 3.3 is applicable. Let ( G , G ) be the separationof G , and let L i be the list-assignment of G i defined in the statement of Lemma 3.3. It iseasily seen that | V ( G i ) | < | V ( G ) | . Thus, by the choice of G , each G i has an ( η, g ) -bounded L i -coloring. Lemma 3.3 then implies that G has an ( η, g ) -bounded L -coloring. Consider a graph with no K s,t -subgraph. In the case s = 1 , a key step in the above proof ofLemma 3.4 uses the fact that if X is a bounded-size set of vertices in a graph G , then N G ( X ) also has bounded size (since G has bounded maximum degree). This fails for graphs with no K s,t -subgraph for s > , since vertices in X might have unbounded degree. To circumventthis issue we focus on those vertices with at least s neighbors in X , and then show that thereare a bounded number of such vertices. The following notation formalizes this simple butimportant idea. For a graph G , a set X ⊆ V ( G ) , and s ∈ N , define N > sG ( X ) := { v ∈ V ( G ) − X : | N G ( v ) ∩ X | > s } and N For all s, t ∈ N , there exists a function f s,t : N → N such that for every graph G with no K s,t -subgraph, if X ⊆ V ( G ) then | N > s ( X ) | f s,t ( | X | ) .Proof. Define f s,t ( x ) := (cid:0) xs (cid:1) ( t − for every x ∈ N . For every y ∈ N > s ( X ) , let Z y be asubset of N G ( y ) ∩ X with size s . Since there are exactly (cid:0) | X | s (cid:1) subsets of X with size s ,if | N > s ( X ) | > f s,t ( | X | ) , then there exists a subset Y of N > s ( X ) with size t such that Z y is identical for all y ∈ Y , implying G [ Y ∪ S y ∈ Y Z y ] contains a K s,t -subgraph, which is acontradiction. Thus | N > s ( X ) | f s,t ( | X | ) .The function f in Lemma 3.5 can be improved if we know more about the graph G ,which helps to get better upper bounds on the clustering. For a graph G , let ∇ ( G ) be themaximum average degree of a graph H for which the 1-subdivision of H is a subgraph of G .The following result is implicit in [55]. We include the proof for completeness. Lemma 3.6. For all s, t ∈ N and ∇ ∈ R + there is a number c := max { t − , ∇ + ( t − (cid:0) ∇ s − (cid:1) } ,such that for every graph G with no K s,t -subgraph and with ∇ ( G ) ∇ , if X ⊆ V ( G ) then | N > s ( X ) | c | X | .Proof. In the case s = 1 , the proof of Lemma 3.5 implies this lemma since c > t − . Nowassume that s > . Let H be the bipartite graph with bipartition { N > sG ( X ) , (cid:0) X (cid:1) } , where v ∈ N > sG ( X ) is adjacent in H to { x, y } ∈ (cid:0) X (cid:1) whenever x, y ∈ N G ( v ) ∩ X . Let M be amaximal matching in H . Let Q be the graph with vertex-set X , where xy ∈ E ( Q ) whenever19 v, { x, y }} ∈ M for some vertex v ∈ N > s ( X ) . Thus, the 1-subdivision of every subgraph of Q is a subgraph of G . Hence | M | = | E ( Q ) | ∇ | V ( Q ) | = ∇ | X | . Moreover, Q is ∇ -degenerate,implying Q contains at most (cid:0) ∇ s − (cid:1) | X | cliques of size exactly s . Exactly | M | vertices in N > sG ( X ) are incident with an edge in M . For each vertex v ∈ N > sG ( X ) not incident with an edge in M , by maximality, N G ( v ) ∩ X is a clique in Q of size at least s . Define a mapping fromeach vertex v ∈ N > sG ( X ) to a clique of size exactly s in Q [ N G ( v ) ∩ X ] . Since G has no K s,t -subgraph, at most t − vertices v ∈ N > sG ( X ) are mapped to each fixed s -clique in Q . Hence | N > sG ( X ) | ∇ | X | + ( t − (cid:0) ∇ s − (cid:1) | X | c | X | .Lemma 3.6 is applicable in many instances. In particular, every graph G with treewidth k has ∇ ( G ) k (since if a 1-subdivision of some graph G ′ is a subgraph of G , then G ′ has treewidth at most k , and every graph with treewidth at most k has average de-gree less than k ). By an analogous argument, using bounds on the average degree in-dependently due to Thomason [73] and Kostochka [42], every H -minor-free graph G has ∇ ( G ) O ( | V ( H ) | p log | V ( H ) | ) . Similarly, using bounds on the average degree independentlydue to Komlós and Szemerédi [41] and Bollobás and Thomason [5], every H -subdivision-freegraph G has ∇ ( G ) O ( | V ( H ) | ) . Finally, Dujmović et al. [17, Lemmas 8,9] proved that ∇ ( G ) w for every graph with layered treewidth w . In all these cases, Lemma 3.6 impliesthat the function f in Lemma 3.5 can be made linear. This improves the clustering functionin many of our results, but for the sake of simplicity, we choose not to explicitly evaluate theclustering functions in these instances. The coloring argument used for graphs of bounded degree in Section 3.1 fails for graphswith no K s,t -subgraph (where s > ) since the precolored set Y might grow too fast whenthe maximum degree is unbounded. We employ the following alternative strategy. Instead ofprecoloring every vertex that is adjacent to Y , only precolor those vertices that are adjacent toat least s vertices in Y , forbidding exactly one color in Y . For each vertex v that is adjacentto at least 1 but at most s − vertices in Y , we ensure (using a list coloring argument)that in the future v is assigned a color that appears on no precolored neighbor of v . Thisallows us to enlarge Y to obtain a larger precolored set Y ′ , such that in every coloring, everymonochromatic component that intersects Y is contained in G [ Y ′ ] , and still | Y ′ | is boundedby a function of | Y | . Lemma 3.5 ensures that the size of the precolored set does not increasetoo much, which is then used to ensure that the final precolored set Y ′ has bounded size.The following type of list assignment formalizes the above intuition, and is used throughoutthe paper. For a graph G , a subset Y ⊆ V ( G ) , and s, r ∈ N , a list-assignment L of G is an ( s, r, Y ) -list-assignment if:(L1) | L ( v ) | ∈ [ s + r ] for every v ∈ V ( G ) .(L2) Y = { v ∈ V ( G ) : | L ( v ) | = 1 } .(L3) For every y ∈ N Theorem 3.7. For all k, s, t ∈ N there exists λ ∈ N , such that every graph with treewidthless than k and with no K s,t -subgraph is ( s + 1) -choosable with clustering λ . Theorem 3.7 is implied by the following lemma, since the s = 1 case is handled byTheorem 3.1, and if L is an ( s + 1) -list assignment, then removing | L ( v ) | − ( s + 1) colors fromeach list L ( v ) gives an ( s, , ∅ ) -list assignment. This is the only place where we assume that s > in this paper. Lemma 3.8. For all s, t, k ∈ N with s > there exists η ∈ N and a nondecreasing function g : N → N such that if G is a graph of treewidth less than k and with no K s,t -subgraph, Y is a subset of V ( G ) with | Y | η , and L is an ( s, , Y ) -list-assignment of G , then there existsan ( η, g ) -bounded L -coloring of G .Proof. Fix the constant c = c ( k, s, t ) from Lemma 3.6 obtained with ∇ = 2 k (since graphswith treewidth less than k have ∇ < k ). Define η := 12 k ( c + 1) k . Define the function g by g ( n ) := ( c + 1) n for n ∈ N .Suppose for the sake of contradiction that there is a counterexample. Choose a coun-terexample ( G, Y, L ) such that | V ( G ) | is minimized, and subject to this condition, | Y | ismaximized.We first deal with two trivial cases. Case A. Y = ∅ : Let x be a vertex of G . Let L ′ ( x ) be a 1-element subset of L ( x ) .For each vertex v ∈ N ( x ) , let L ′ ( v ) be a subset of L ( v ) − L ′ ( x ) of size s . For each vertex v ∈ V ( G ) − N [ x ] , let L ′ ( v ) := L ( v ) . Let Y ′ := { x } . Then L ′ is an ( s, , Y ′ ) -list assignmentfor G . By the choice of counterexample, ( G, L ′ ) is not a counterexample. Thus there existsan ( η, g ) -bounded L ′ -coloring of G , which is also an ( η, g ) -bounded L -coloring of G . Hence ( G, L ) is not a counterexample. Now assume that Y = ∅ . Case B. N ( Y ) = ∅ : By the choice of counterexample, ( G − Y, ∅ , L ) is not a counterex-ample. Thus G − Y has an ( η, g ) -bounded L -coloring. Assign each vertex v ∈ Y the colorin L ( v ) . Since N ( Y ) = ∅ , every monochromatic component intersecting Y is contained in G [ Y ] , which has size | Y | | Y | g ( | Y | ) . Thus we have an ( η, g ) -bounded L -coloring of G . Nowassume that N ( Y ) = ∅ .We now deal with the two main cases. Case C. | Y | ∈ [1 , k − : Say Y =: { x , . . . , x | Y | } . Let Y := Y and L := L . Define Y , L , . . . , Y | Y | , L | Y | recursively as follows. For i = 1 , , . . . , | Y | ,21 for v ∈ N > s ( Y i − ) , let L i ( v ) be a 1-element subset of L i − ( v ) − L ( x i ) , • let Y i := Y i − ∪ N > s ( Y i − ) , • for v ∈ N For each i ∈ [0 , | Y | ] , L i is an ( s, , Y i ) -list-assignment for G and | Y i | ( c +1) i | Y | .Proof. We proceed by induction on i > . By assumption L = L is an ( s, , Y ) -list assign-ment and | Y | = | Y | = ( c + 1) | Y | . Now assume that i > , and that L i − is an ( s, , Y i − ) -listassignment for G .First, we show that L i is well-defined. Say v ∈ N > s ( Y i − ) . Then v Y i − , implying | L i − ( v ) | > . Moreover, | L ( x i ) | = 1 . Thus L i − ( v ) − L ( x i ) = ∅ , implying L i ( v ) is well-defined. Now, consider v ∈ N 1) = 2 . If v ( Y i ∪ N Lemma 4.1 ([1]) . There is a function f : N × N → N such that every graph with treewidth w and maximum degree ∆ is 2-colorable with clustering f ( w, ∆) ∈ O ( k ∆) . For a graph G with bounded maximum degree and bounded layered treewidth, if ( V , . . . , V n ) is the corresponding layering of G , then Lemma 4.1 is applicable to G [ V i ] , whichhas bounded treewidth. The idea of the proof of Theorem 1.18 is to use colors 1 and 2 for all23ayers V i with i ≡ , use colors 2 and 3 for all layers V i with i ≡ , and usecolors 3 and 1 for all layers V i with i ≡ . Then each monochromatic componentis contained within two consecutive layers. The key to the proof is to control the growth ofmonochromatic components between consecutive layers. The next lemma is useful for thispurpose. Lemma 4.2. Let w, ∆ , d, k, h ∈ N . Let G be a graph with maximum degree at most ∆ . Let ( T, X ) be a tree-decomposition of G with width at most w , where X = ( X t : t ∈ V ( T )) .For i > , let Y i be a subset of V ( T ) , T i a subtree of T containing Y i , and E i a set ofpairs of vertices in S x ∈ Y i X x with | E i | k . Let G ′ be the graph with V ( G ′ ) = V ( G ) and E ( G ′ ) = E ( G ) ∪ S i > E i . If every vertex of G appears in at most d pairs in S i > E i , andevery vertex of T is contained in at most h members of { T , T , . . . } , then G ′ has maximumdegree at most ∆ + d and has a tree-decomposition ( T, X ′ ) of width at most w + 2 hk .Proof. Since every vertex of G appears in at most d pairs in S i > E i , G ′ has maximumdegree at most ∆ + d . For every i > , let Z i be the set of the vertices appearing in somepair of E i . Note that Z i ⊆ S x ∈ Y i X x and | Z i | | E i | k . For every t ∈ V ( T ) , let X ′ t := X t ∪ S { i : t ∈ V ( T i ) } Z i . Let X ′ := ( X ′ t : t ∈ V ( T )) .We claim that ( T, X ′ ) is a tree-decomposition of G ′ . It is clear that S t ∈ V ( T ) X ′ t ⊇ V ( G ′ ) .For each i ∈ N , for every t ∈ V ( T i ) , since X ′ t ⊇ Z i , X ′ t contains both ends of each edge in E i .For each v ∈ V ( G ′ ) , { t ∈ V ( T ) : v ∈ X ′ t } = { t ∈ V ( T ) : v ∈ X t } ∪ [ { i : v ∈ Z i } V ( T i ) . Note that for every v ∈ V ( G ) and i > , if v ∈ Z i then v ∈ X t for some t ∈ Y i ⊆ V ( T i ) .Hence { t : v ∈ X ′ t } induces a subtree of T . This proves that ( T, X ′ ) is a tree-decompositionof G ′ .Since for every t ∈ V ( T ) , | X ′ t | | X t | + P { i : t ∈ V ( T i ) } | Z i | w + 1 + 2 hk , the width of ( T, X ′ ) is at most w + 2 hk .We now prove Theorem 1.18. Theorem 4.3. Let ∆ , w ∈ N . Then every graph G with maximum degree at most ∆ andwith layered treewidth at most w is 3-colorable with clustering g ( w, ∆) , for some function g ( w, ∆) ∈ O ( w ∆ ) .Proof. Let f be the function from Lemma 4.1. Define f := f ( w, ∆) ∈ O ( w ∆)∆ := ∆ + f ∆ ∈ O ( w ∆ ) w := w + 2( w + 1) f ∆ ∈ O ( w ∆ ) ,f := f ( w , ∆ ) ∈ O ( w ∆ )∆ := ∆ + f ∆ ∈ O ( w ∆ ) w := w + 4( w + 1) f ∆ ∈ O ( w ∆ ) f := f ( w , ∆ ) ∈ O ( w ∆ ) ( w, ∆) := (1 + f ∆) f ∈ O ( w ∆ ) . Let G be a graph of maximum degree at most ∆ and layered treewidth at most w . Let ( T, X ) and ( V i : i > be a tree-decomposition of G and a layering of G such that | X t ∩ V i | w for every t ∈ V ( T ) and i > , where X = ( X t : t ∈ V ( T )) . For j ∈ [3] , let U j = S ∞ i =0 V i + j .By Lemma 4.1, there exists a coloring c : U → { , } such that every monochromaticcomponent of G [ U ] contains at most f vertices. For each i ∈ N , let C i be the set of c -monochromatic components of G [ U ] contained in V i +1 with color 2. For each i ∈ N and C ∈ C i , define the following: • Let Y i,C be a minimal subset of V ( T ) such that for every edge e of G between V ( C ) and N G ( V ( C )) ∩ V i +2 , there exists a node t ∈ Y i,C such that both ends of e belong to X t . • Let E i,C be the set of all pairs of distinct vertices in N G ( V ( C )) ∩ V i +2 . • Let T i,C be the subtree of T induced by { t ∈ V ( T ) : X t ∩ V ( C ) = ∅} .Note that there are at most | V ( C ) | ∆ f ∆ edges of G between V ( C ) and N G ( V ( C )) . So | Y i,C | f ∆ and | E i,C | f ∆ for every i ∈ N and C ∈ C i . In addition, Y i,C ⊆ V ( T i,C ) forevery i ∈ N and C ∈ C i . Since ( T, ( X t ∩ U : t ∈ V ( T ))) is a tree-decomposition of G [ U ] withwidth at most w , for every t ∈ V ( T ) and i ∈ N , there exist at most w + 1 different members C of C i such that t ∈ V ( T i,C ) . Furthermore, N G ( V ( C )) ∩ V i +2 ⊆ S x ∈ Y i,C X x , so each pair in E i,C consists of two vertices in S x ∈ Y i,C X x . Since every vertex v in U is adjacent in G to atmost ∆ members of S i ∈ N C i and every member C of S i ∈ N C i creates at most f ∆ pairs in E i C ,C involving v , where i C is the index such that C ∈ C i C , every vertex in U appears in atmost f ∆ · ∆ = f ∆ pairs in S i ∈ N ,C ∈C i E i,C .Let G be the graph with V ( G :) = U and E ( G ) := E ( G [ U ]) ∪ [ i ∈ N ,C ∈C i E i,C . We have shown that Lemma 4.2 is applicable with k = f ∆ and d = f ∆ and h = w + 1 .Thus G has maximum degree at most ∆ and a tree-decomposition ( T, X (2) ) of width at most w . Say X (2) = ( X (2) t : t ∈ V ( T )) . By Lemma 4.1, there exists a coloring c : U → { , } such that every c -monochromatic component of G contains at most f vertices.Note that we may assume that ( T, X (2) ) is a tree-decomposition of G [ U ∪ U ] by redefining X (2) t to be the union of X (2) t and X t ∩ U , for every t ∈ V ( T ) .For each i ∈ N , let C ′ i be the set of the c -monochromatic components of G [ U ] with color1 and the c -monochromatic components of G with color 3. For each i ∈ N and C ∈ C ′ i ,define the following: • Let Y ′ i,C be a minimal subset of V ( T ) such that for every edge e of G between V ( C ) and N G ( V ( C )) ∩ V i +3 , there exists a node t ∈ Y ′ i,C such that both ends of e belong to X t . • Let E ′ i,C be the set of all pairs of distinct vertices of N G ( V ( C )) ∩ V i +3 . • Let T ′ i,C be the subtree of T induced by { t ∈ V ( T ) : X (2) t ∩ V ( C ) = ∅} .Note that there are at most | V ( C ) | ∆ f ∆ edges of G between V ( C ) and N G ( V ( C )) ∩ V i +3 for every i ∈ N and C ∈ C ′ i . So | Y ′ i,C | f ∆ and | E ′ i,C | f ∆ for every i ∈ N and C ∈ C ′ i .25n addition, Y ′ i,C ⊆ V ( T ′ i,C ) for every i ∈ N and C ∈ C ′ i . Since ( T, ( X (2) t ∩ U : t ∈ V ( T ))) is a tree-decomposition of G with width at most w and ( T, ( X (2) t ∩ U : t ∈ V ( T ))) is atree-decomposition of G [ U ] with width at most w w , for every t ∈ V ( T ) and i ∈ N , thereexist at most w + 1) different members C ∈ C i such that t ∈ V ( T ′ i,C ) . Furthermore, eachpair in E ′ i,C consists of two vertices in S x ∈ Y ′ i,C X (2) x . Since every vertex v in U is adjacent in G to at most ∆ members of S i ∈ N C ′ i , and every member C of S i ∈ N C ′ i creates at most f ∆ pairs in E i C ,C involving v , where i C is the index such that C ∈ C ′ i C , every vertex in U appearsin at most f ∆ pairs in S i ∈ N ,C ∈C ′ i E i,C .Let G be the graph with V ( G ) := U and E ( G ) := E ( G [ U ]) ∪ [ i ∈ N ,C ∈C ′ i E ′ i,C . We have shown that Lemma 4.2 is applicable with k = f ∆ and d = f ∆ and h = 2( w +1) . Hence G has maximum degree at most ∆ and a tree-decomposition ( T, X (3) ) withwidth at most w . By Lemma 4.1, there exists a coloring c : U → { , } such that everymonochromatic component of G contains at most f vertices.Define c : V ( G ) → { , , } such that for every v ∈ V ( G ) , we have c ( v ) := c j ( v ) , where j is the index for which v ∈ U j . Now we prove that every c -monochromatic component of G contains at most g ( w, ∆) vertices.Let D be a c -monochromatic component of G with color 2. Since D is connected, forevery pair of vertices u, v ∈ V ( D ) ∩ U , there exists a path P uv in D from u to v . Since V ( D ) ⊆ U ∪ U , for every maximal subpath P of P uv contained in U , there exists i ∈ N and C ∈ C i such that there exists a pair in E i,C consisting of the two vertices in P adjacent tothe ends of P uv . That is, there exists a path in G [ V ( D ) ∩ U ] connecting u, v for every pair u, v ∈ V ( D ) ∩ U . Hence G [ V ( D ) ∩ U ] is connected. So G [ V ( D ) ∩ U ] is a c -monochromaticcomponent of G with color 2 and contains at most f vertices. Hence there are at most f ∆ edges of G between V ( D ) ∩ U and N G ( V ( D ) ∩ U ) . So D [ V ( D ) ∩ U ] contains at most f ∆ components. Since each component of D [ V ( D ) ∩ U ] is a c -monochromatic component of G [ U ] , it contains at most f vertices. Hence D [ V ( D ) ∩ U ] contains at most f ∆ f vertices.Since D has color 2, V ( D ) ∩ U = ∅ . Therefore, D contains at most (1 + f ∆) f g ( w, ∆) vertices.Let D ′ be a c -monochromatic component of G with color b , where b ∈ { , } . Since D ′ is connected, by an analogous argument to that in the previous paragraph, G [ V ( D ′ ) ∩ U ] is connected. So G [ V ( D ′ ) ∩ U ] is a c -monochromatic component of G with color b andcontains at most f vertices. Hence there are at most f ∆ edges of G between V ( D ′ ) ∩ U and N G ( V ( D ′ ) ∩ U ) . So D [ V ( D ) ∩ U b ′ ] contains at most f ∆ components, where b ′ = 1 if b = 1 and b ′ = 2 if b = 3 . Since each component of D [ V ( D ) ∩ U b ′ ] is a c b ′ -monochromaticcomponent of G [ U b ′ ] , it contains at most f vertices. Hence D [ V ( D ) ∩ U b ′ ] contains at most f ∆ f vertices. Since D has color b , V ( D ) ∩ U b +1 = ∅ , where U = U . Therefore, D containsat most (1 + f ∆) f g ( w, ∆) vertices. This completes the proof. The next definition formalizes and generalizes the idea in Case C of the proof of Lemma 3.8for graphs of bounded treewidth, and is used throughout the remainder of the paper. Let26 be a graph, let Y ⊆ V ( G ) , let s, r ∈ N , and let L be an ( s, r, Y ) -list-assignment. Forall W ⊆ V ( G ) and for every set F of colors with | F | r (not necessarily a subset of S v ∈ V ( G ) L ( v ) ), a ( W, F ) -progress of L is a list-assignment L ′ of G defined as follows: • Let Y ′ := Y ∪ W . • For every y ∈ Y , let L ′ ( y ) := L ( y ) . • For every y ∈ Y ′ − Y , let L ′ ( y ) be a 1-element subset of L ( y ) − F (which exists by(L5)). • For each v ∈ N Let G be a graph, s, r ∈ N , and L be an ( s, r, Y ) -list-assignment. If W ⊆ V ( G ) and F is a set of colors with | F | r , then every ( W, F ) -progress L ′ of L satisfies the followingproperties:1. L ′ is an ( s, r, Y ∪ W ) -list-assignment of G .2. L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) .3. { v ∈ Y ∪ W : L ′ ( v ) ∩ F = ∅} = { v ∈ Y : L ( v ) ∩ F = ∅} .4. If N > s ( Y ) ⊆ W , then for every y ∈ Y ∪ W and color x ∈ F ∩ L ′ ( y ) , we have { v ∈ N G ( y ) − ( Y ∪ W ) : x ∈ L ′ ( v ) } = ∅ .5. For every v ∈ V ( G ) − ( Y ∪ W ) , we have L ′ ( v ) ∩ F = L ( v ) ∩ F .Proof. Let L ′ be a ( W, F ) -progress of L . By construction, Statements 2 and 3 hold. Let Y ′ := Y ∪ W .Now we prove Statement 4. Suppose to the contrary that there exist y ∈ Y ′ , b ∈ N G ( y ) − Y ′ and f ∈ F ∩ L ′ ( b ) ∩ L ′ ( y ) . Since f L ′ ( q ) for every q ∈ Y ′ − Y , we have y ∈ Y . Since N > s ( Y ) ⊆ W and b W , we have | N G ( b ) ∩ Y | ∈ [ s − . That is, b ∈ N Let G be a graph and let s ∈ N . Let V = ( V , V , . . . ) be a layering of G ,and let L be an ( s, V ) -compatible list-assignment. If k ∈ N and c is an L -coloring such thatfor every i ∈ [ s + 2] and every s -segment S with level i , every c -monochromatic componentcontained in G [ S ] with color i contains at most k vertices, then c has clustering at most k .Proof. Since for every i ∈ [ s + 2] , if v ∈ V ( G ) with i ∈ L ( v ) , then v belongs to a segment of V with level i , so every c -monochromatic component with color i is contained in some segmentof V with level i .Let G be a graph, Z ⊆ V ( G ) , s a positive integer, V a Z -layering of G , and L an ( s, V ) -compatible list-assignment of G . For Y ⊆ V ( G ) , we say that ( Y , L ) is a V -standard pair if L is an ( s, V ) -compatible ( s, , Y ) -list-assignment of G ’.Note that if L is ( s, V ) -compatible and V = ( V , V , . . . ) , then for each vertex v ∈ V j with j ≡ i (mod s + 2) , we have L ( v ) ⊆ [ s + 2] − { i } , implying that | L ( v ) | ∈ [ s + 1] , which iscondition (L1) in an ( s, , Y ) -list-assignment.For a color i , if L ′ is a ( W, { i } ) -progress of L , then the pair ( Y ′ , L ′ ) is called a ( W, i ) -progress. So in a ( W, i ) -progress, vertices in W − Y become precolored by a color differentfrom i . We implicitly use the following obvious observation throughout Section 4. Observation 4.6. Let G be a graph, let V be a Z -layering for some Z ⊆ V ( G ) , and let ( Y , L ) be a V -standard pair. For every set W ⊆ V ( G ) and color i (not necessarily belongingto S v ∈ V ( G ) L ( v ) ), a ( W, i ) -progress is well-defined and is a V -standard pair. This section introduces the notion of a fence, which is used throughout the proof ofTheorem 1.19. We start with a variant of a well-known result about separators in graphs ofbounded treewidth. Lemma 4.7. Let w ∈ N . Let G be a graph and ( T, X ) a tree-decomposition of G of width atmost w , where X = ( X t : t ∈ V ( T )) . If Q is a subset of V ( G ) with | Q | > w + 13 , then thereexists t ∗ ∈ V ( T ) such that for every component T ′ of T − t ∗ , | ( Q ∩ ( S t ∈ V ( T ′ ) X t )) ∪ X t ∗ | < | Q | .Proof. Suppose that there exists an edge xy of T such that | ( Q ∩ ( S t ∈ V ( T i ) X t )) ∪ X x ∪ X y | > | Q | for each i ∈ [2] , where T , T are the components of T − xy . Then | Q | + 2 | X x ∪ X y | > X i =1 | ( Q ∩ ( [ t ∈ V ( T i ) X t )) ∪ X x ∪ X y | > | Q | . Since the width of ( T, X ) is at most w , w +1) > | X x ∪ X y | > | Q | > w + , a contradiction.29irst assume that there exists an edge xy of T such that | ( Q ∩ ( S t ∈ V ( T i ) X t )) ∪ X x ∪ X y | < | Q | for each i ∈ [2] , where T , T are the components of T − xy . Let t ∗ := x . Then for everycomponent T ′ of T − t ∗ , ( Q ∩ ( S t ∈ V ( T ′ ) X t )) ∪ X t ∗ ⊆ ( Q ∩ ( S t ∈ V ( T i ) X t )) ∪ X x ∪ X y for some i ∈ [2] , and hence | ( Q ∩ ( S t ∈ V ( T ′ ) X t )) ∪ X t ∗ | < | Q | . So the lemma holds.Now assume that for every edge xy of T , there exists a unique r ∈ { x, y } such that | ( Q ∩ ( S t ∈ V ( T r ) X t )) ∪ X x ∪ X y | > | Q | , where T x , T y are the components of T − xy containing x, y , respectively. Orient the edge xy so that r is the head of this edge. We obtain anorientation of T . Note that the sum of the out-degree of the nodes of T equals | E ( T ) | = | V ( T ) | − . So some node t ∗ has out-degree 0.For each component T ′ of T − t ∗ , let t T ′ be the node in T ′ adjacent in T to t ∗ . By thedefinition of the direction of t T ′ t ∗ , | ( Q ∩ ( [ t ∈ V ( T tT ′ ) X t )) ∪ X t ∗ | | ( Q ∩ ( [ t ∈ V ( T tT ′ ) X t )) ∪ X t T ′ ∪ X t ∗ | < | Q | . Note that T t T ′ = T ′ for every component T ′ of T − t ∗ . This proves the lemma.Let T be a tree and F a subset of V ( T ) . An F -part of T is an induced subtree of T obtainedfrom a component of T − F by the adding nodes in F adjacent in T to this component. Foran F -part T ′ of T , define ∂T ′ to be F ∩ V ( T ′ ) . Lemma 4.8. Let ( T, X ) be a tree-decomposition of a graph G of width at most w ∈ N , where X = ( X t : t ∈ V ( T )) . Then for every ǫ ∈ R with > ǫ > w +1 and every Q ⊆ V ( G ) , thereexists a subset F of V ( T ) with | F | max { ǫ ( | Q | − w − , } such that for every F -part T ′ of T , | ( S t ∈ V ( T ′ ) X t ∩ Q ) ∪ S t ∈ ∂T ′ X t | ǫ (12 w + 13) .Proof. We shall prove this lemma by induction on | Q | . If | Q | ǫ (12 w + 13) , then let F := ∅ ;then for every F -part T ′ of T , | ( S t ∈ V ( T ′ ) X t ∩ Q ) ∪ S t ∈ ∂T ′ X t | = | Q | ǫ (12 w + 13) . So wemay assume that | Q | > ǫ (12 w + 13) > w + 13 and the lemma holds for all Q with smallersize.By Lemma 4.7, there exists t ∗ ∈ V ( T ) such that for every component T ′ of T − t ∗ , | ( Q ∩ ( S t ∈ V ( T ′ ) X t )) ∪ X t ∗ | < | Q | . That is, for every { t ∗ } -part T ′ of T , | ( S t ∈ V ( T ′ ) X t ∩ Q ) ∪ S t ∈ ∂T ′ X t | < | Q | .For each { t ∗ } -part T ′ of T , let Q T ′ := ( S t ∈ V ( T ′ ) X t ∩ Q ) ∪ X t ∗ , so | Q T ′ | < | Q | . Let T , T , . . . , T k be the { t ∗ } -parts T ′ of T with | Q T ′ | > ǫ (12 w + 13) .Let f be the function defined by f ( x ) := max { ǫ ( x − w − , } for every x ∈ R . For each i ∈ [ k ] , since | Q T i | < | Q | < | Q | , the induction hypothesis implies that there exists F i ⊆ V ( T i ) with | F i | f ( | Q T i | ) such that for every F i -part T ′ of T i , | ( S t ∈ V ( T ′ ) X t ∩ Q T i ) ∪ S t ∈ ∂T ′ X t | ǫ (12 w + 13) .Define F = { t ∗ } ∪ S ki =1 F i . Note that for every F -part T ′ of T , either T ′ is a { t ∗ } -partof T with | Q T ′ | ǫ (12 w + 13) , or there exists i ∈ [ k ] such that T ′ is an F i -part of T i .In the former case, | ( S t ∈ V ( T ′ ) X t ∩ Q ) ∪ S t ∈ ∂T ′ X t | = | Q T ′ | ǫ (12 w + 13) . In the lattercase, | ( S t ∈ V ( T ′ ) X t ∩ Q ) ∪ S t ∈ ∂T ′ X t | | ( S t ∈ V ( T ′ ) X t ∩ Q T i ) ∪ S t ∈ ∂T ′ X t | ǫ (12 w + 13) since X t ∗ ⊆ Q T i . Hence | ( S t ∈ V ( T ′ ) X t ∩ Q ) ∪ S t ∈ ∂T ′ X t | ǫ (12 w + 13) for every F -part T ′ of T .30o prove this lemma, it suffices to prove that | F | f ( | Q | ) . Note that | F | P ki =1 | F i | P ki =1 f ( | Q T i | ) . Since | Q T i | > ǫ (12 w + 13) > w + 13 for every i ∈ [ k ] , | F | P ki =1 ǫ ( | Q T i | − w − .If k = 0 , then f ( | Q | ) = ǫ ( | Q | − w − 3) = ǫ | Q | − ǫ (3 w + 3) > w + 13 − (3 w + 3) > | F | since ǫ | Q | > w + 13 and ǫ . If k = 1 , then | F | ǫ ( | Q T | − w − ǫ ( | Q | − w − 3) = ǫ ( | Q | − w − 3) + 1 − ǫ | Q | f ( | Q | ) since ǫ | Q | > w + 13 . Hence we may assumethat k > . Then | F | k X i =1 ǫ ( | Q ∩ [ t ∈ V ( T i ) −{ t ∗ } X t | + | X t ∗ | − w − ǫ | Q | + kǫ ( | X t ∗ | − w − ǫ ( | Q | − w − 3) + 1 + kǫ | X t ∗ | − ( k − ǫ (3 w + 3) f ( | Q | ) + 1 + ǫ ( k ( w + 1) − k − w + 1)) f ( | Q | ) since k > . This proves the lemma. Lemma 4.9. Let ( T, X ) be a tree-decomposition of a graph G of width at most w ∈ N , where X = ( X t : t ∈ V ( T )) . Then for every Q ⊆ V ( G ) , there exists a subset F of V ( T ) with | F | max {| Q | − w − , } such that:1. for every F -part T ′ of T , | S t ∈ V ( T ′ ) X t ∩ Q | w + 13 , and2. if Q = ∅ , then for every t ∗ ∈ F , there exists at least two F -parts T ′ of T satisfying t ∗ ∈ ∂T ′ and Q ∩ S t ∈ V ( T ′ ) X t − X t ∗ = ∅ .Proof. Let Q ⊆ V ( G ) . By Lemma 4.8 with ǫ = 1 , there exists F ⊆ V ( T ) with | F | max {| Q | − w − , } such that for every F -part T ′ of T , | S t ∈ V ( T ′ ) X t ∩ Q | w + 13 .Assume further that F is minimal.Suppose that Q = ∅ and there exists t ∗ ∈ F such that there is at most one F -part T ′ of T satisfying t ∗ ∈ ∂T ′ and Q ∩ S t ∈ ∂T ′ X t − X t ∗ = ∅ . Let F ∗ := F − { t ∗ } . Note that for every F -part W of T with t ∗ ∂W , W is an F ∗ -part of T and ∂W ⊆ F −{ t ∗ } , so | S t ∈ V ( W ) X t ∩ Q | w +13 . In addition, there is exactly one F ∗ -part T ∗ of T with t ∗ ∈ V ( T ∗ ) , and every F ∗ -partof T other than T ∗ is an F -part of T . Since there is at most one F -part T ′ of T satisfying t ∗ ∈ ∂T ′ and Q ∩ S t ∈ ∂T ′ X t − X t ∗ = ∅ , we know | S t ∈ V ( T ∗ ) X t ∩ Q | = | S t ∈ V ( T ′ ) X t ∩ Q | w +13 .This contradicts the minimality of F and proves the lemma.We call the set F mentioned in Lemma 4.9 a ( T, X , Q ) -fence . This subsection introduces parades and fans, and proves an auxiliary lemma that will beused in the proof of Theorem 1.17.Let ( T, X ) be a rooted tree-decomposition of a graph G of width w , where X = ( X t : t ∈ V ( T )) . Let t, t ′ be distinct nodes of T with t ′ ∈ V ( T t ) . Let T t be the subtree of T rooted at t .For k ∈ [0 , w + 1] and m ∈ N , a ( t, t ′ , k ) -fan of size m is a sequence ( t , t , . . . , t m ) of nodessuch that: 31FAN1) for every j ∈ [ m − , t j +1 ∈ V ( T t j ) − { t j } and t ′ ∈ V ( T t j ) ,(FAN2) for every j ∈ [ m ] , | X t j ∩ X t | = k , and(FAN3) X t j − X t are pairwise disjoint for all j ∈ [ m ] .Let T be a rooted tree. A parade in T is a sequence ( t , t , . . . , t k ) of nodes of T (for some k ∈ N ) such that for every α ∈ [ k − , t α +1 ∈ V ( T t α ) − { t α } .A sequence ( a , a , . . . , a α ) is a subsequence of a sequence ( b , b , . . . , b β ) (for some positiveintegers α, β ) if there exists an injection ι : [ α ] → [ β ] such that ι (1) < ι (2) < · · · < ι ( α ) and a i = b ι ( i ) for every i ∈ [ α ] . Lemma 4.10. For every w ∈ N and k ∈ N , there exists N := N ( w, k ) ∈ N such that forevery rooted tree-decomposition ( T, X ) of a graph G , where X = ( X t : t ∈ V ( T )) , and forevery parade ( t , t , . . . , t N ) in T with | X t i | w + 1 for every i ∈ [ N ] , if X t i X t j forevery i > j , then there exist ℓ ∈ [0 , w ] and a ( t ′ , t ′ k , ℓ ) -fan ( t ′ , t ′ , . . . , t ′ k ) of size k that is asubsequence of ( t , t , . . . , t N ) .Proof. Define N (0 , k ) := max { k, } , and for every x ∈ [ w ] , define N ( x, k ) := N ( x − , k ) · ( k − w + 1) .We proceed by induction on w . When w = 0 , since X t i X t j for every i > j , ( t , t , . . . , t N ) is a ( t , t N , -fan of size N > k . So we may assume that w > and this lemma holds forevery smaller w .Suppose to the contrary that for every ℓ ∈ [0 , w ] , there exists no ( t ′ , t ′ k , ℓ ) -fan ( t ′ , t ′ , . . . , t ′ k ) of size k that is a subsequence of ( t , t , . . . , t N ) . In particular, there exists no k elements in { t , t , . . . , t k } with disjoint bags. Hence there exists a parade ( q , q , . . . , q k − ) that is asubsequence of ( t , t , . . . , t N ) such that for every i ∈ [ N ] , there exists i ′ ∈ [ k − suchthat X t i ∩ X q i ′ = ∅ and t i ∈ V ( T q i ′ ) . For each i ∈ [ k − , let S i := { t j ∈ V ( T q i ) : j ∈ [ N ] , X t j ∩ X q i = ∅} . So there exists i ∈ [ k − such that | S i | > N/ ( k − . For every α ∈ [ w + 1] , let S ′ α := { t j ∈ S i : | X t j ∩ X q i | = α } . So there exists α ∗ ∈ [ w + 1] such that | S ′ α ∗ | > | S i | / ( w + 1) > N ( k − w +1) > . Since | X t i | w + 1 for every i ∈ [ N ] , and X t i X t j for every i > j , α ∗ ∈ [ w ] .Let ( z , z , . . . , z | S ′ α ∗ | ) be the parade formed by the element of S ′ α ∗ . Since ( T, X ) is atree-decomposition, there exists Z ⊆ X q i with | Z | = α ∗ such that for every i ∈ [ | S ′ α ∗ | ] , X z i ∩ X q i = Z .For every t ∈ V ( T ) , let X ′ t := X t − Z . Let X ′ := ( X ′ t : t ∈ V ( T )) . Then | X ′ z i | = | X z i | − | Z | ( w − 1) + 1 for every i ∈ [ | S ′ α ∗ | ] . If there exist i, j ∈ [ | S ′ α ∗ | ] with i > j such that X ′ z i ⊆ X ′ z j , then X z i = X ′ z i ∪ Z ⊆ X ′ z j ∪ Z = X z j , a contradiction. So for any i, j ∈ [ | S ′ α ∗ | ] with i > j , X ′ z i X ′ z j . Since | S ′ α ∗ | > N ( k − w +1) > N ( w − , k ) , by the induction hypothesis, thereexist ℓ ∈ [0 , w − | Z | ] and a ( t ′ , t ′ k , ℓ ) -fan ( t ′ , t ′ , . . . , t ′ k ) in ( T, X ′ ) of size k that is a subsequenceof ( z , z , . . . , z | S ′ α ∗ | ) and hence is a subsequence of ( t , t , . . . , t N ) . Since Z ⊆ X z i for every i ∈ [ | S ′ α ∗ | ] , ( t ′ , t ′ , . . . , t ′ k ) is a ( t ′ , t ′ k , ℓ + | Z | ) -fan in ( T, X ) . Note that ℓ + | Z | ∈ [0 , w ] . Thisproves the lemma. The next lemma implies Theorem 1.17 by taking η = 0 and Y = ∅ . Most of the workin proving Lemma 4.11 is done by Lemma 4.12. So we prove Lemma 4.11 first, assumingLemma 4.12. 32 emma 4.11. For every s, t, w ∈ N and η ∈ N , there exists λ ∈ N such that if G is a graphwith no K s,t -subgraph, V is a layering of G , L is an ( s, V ) -compatible list-assignment, Y isa subset of V ( G ) , ( Y , L ) is a V -standard pair, and ( T, X ) is a tree-decomposition of G of V -width at most w such that | Y ∩ S | η for every s -segment S of V , then there exists an L -coloring of G with clustering λ .Proof of Lemma 4.11 assuming Lemma 4.12. Let s, t, w ∈ N and η ∈ N . Let λ be the num-ber λ ( s, t, w + η ) from Lemma 4.12. Let G be a graph with no K s,t -subgraph, V a layeringof G , L an ( s, V ) -compatible list-assignment, Y a subset of V ( G ) , ( Y , L ) a V -standard pair,and ( T, X ) a tree-decomposition of G of V -width at most w such that | Y ∩ S | η for every s -segment S of V . Say X = ( X t : t ∈ V ( T )) .For each t ∈ V ( T ) , let X ∗ t := X t ∪ Y . Let X ∗ = ( X ∗ t : t ∈ V ( T )) . Let t ∗ be a nodeof T . Then ( T, X ∗ ) is a tree-decomposition of G such that X ∗ t ∗ contains Y . Since for every s -segment S of V , | Y ∩ S | η , the V -width of ( T, X ∗ ) is at most w + η . Therefore, byLemma 4.12, there exists an L -coloring of G with clustering λ .The next lemma is the heart of the proof of Theorem 1.17. Let ( Y , L ) be a V -standardpair of a graph G . For y ∈ Y , a gate for y (with respect to ( Y , L ) ) is a vertex v ∈ N G ( y ) − Y such that L ( v ) ∩ L ( y ) = ∅ . For W ⊆ Y , let A ( Y ,L ) ( W ) := { v ∈ V ( G ) − Y : v is a gate for some y ∈ W with respect to ( Y , L ) } . Lemma 4.12. For every s, t, w ∈ N , there exists λ ∈ N such that if G is a graph with no K s,t -subgraph, V is a layering of G , L is an ( s, V ) -compatible list-assignment, Y is a subsetof V ( G ) , ( Y , L ) is a V -standard pair, and ( T, X ) is a tree-decomposition of G of V -width atmost w such that some bag contains Y , then there exists an L -coloring of G with clustering λ .Proof. Let s, t, w ∈ N . We start by defining several values used throughout the proof: • Let f be the function f s,t,w in Lemma 3.6. • Let f be the identity function f on N ; for every i > , let f be the function from N to N such that f i ( x ) = f i − ( x ) + f ( f i − ( x )) for every x ∈ N . • Let s ∗ := 12( s + 2) . • Let w := 12 w s ∗ + 13 . • Let g : N → N be the function defined by g (0) := w and g ( x ) := f ( g ( x − w for every x ∈ N . • Let g : N → N be the function defined by g (0) := g ( w ) and g ( x ) := f ( g ( x − 1) +3 w ) for every x ∈ N . • Let g : N × N → N be the function defined by g (0 , y ) := y + w for every y ∈ N ,and g ( x, y ) := f ( g ( x − , y )) + 3 w for every x ∈ N . • Let g : N → N be the function defined by g (0) := g ( s + 2) , and g ( x ) := g ( s +2 , g ( x − for every x ∈ N . • Let η := w g ( s + 2) + 3 w g (4 w ) + w . • Let g : N × N → N be the function defined by g (0 , y ) := y and g ( x, y ) := ( s + 2) · f ( g ( x − , y ) + 2 η ) for every x ∈ N and y ∈ N ,33 Let g : N → N be the function defined by g ( x ) := g ( s + 2 , g ( w , x )) + 2 g (4 w ) forevery x ∈ N . • Let g : N → N be the function defined by g (0) := w , and g ( x ) := g ( g ( x − w for every x ∈ N . • Let η := g (8 w ) . • Let η := ( η + 2 η ) w . • Let g : N × N → N be the function defined by g (0 , y ) := y and g ( x, y ) := f ( g ( x − , y )) + w for every x ∈ N and y ∈ N . • Let g : N → N be the function defined by g (0) := η + 2 η and g ( x ) := g ( s +2 , g ( x − for every x ∈ N . • Let g : N → N be the function defined by g (0) := 2 g ( w ) + η and g ( x ) := f ( g ( x − w for every x ∈ N . • Let η := g ( s + 2) . • Let g : N → N be the function defined by g (0) := 2 η and g ( x ) := f s +3 ( g ( x − 1) + η + 2 η ) for every x ∈ N . • Let η := 3 g ( η ) . • Let ψ : [ w ] → N be the function such that for every ( x , x ) ∈ [ w ] , ψ ( x , x ) :=(10( f w ( η )) w +3 w + w ) · (min { x , x } · ( w + 1) · ( f ( η ) + 1) w +1 + 1) . • Let ψ : [ w ] → N be the function such that for every ( x , x ) ∈ [ w ] , ψ ( x , x ) := ψ (100 x , x ) . • Let ψ : [ w ] → N be the function such that for every ( x , x ) ∈ [ w ] , ψ ( x , x ) := ψ ( x , x ) · ( f ( η )) w ψ (0 , . • Let κ := ψ ( w , w ) + ψ ( w , w ) . • Let κ := w · N . w , κ + w + 3) , where N . is the number N mentioned inLemma 4.10. • Let φ : N → N be the function defined by φ (1) := κ , and φ ( x + 1) := κ · (1 + P xi =1 φ ( i )) + 1 for every x ∈ N . • Let φ : N → N be the function defined by φ (0) := 0 , and φ ( x ) := (( φ ( x ) +1) w f ( η ) + 1) · (1 + P x − i =1 φ ( i )) for every x ∈ N . • Let φ : N → N be the function defined by φ ( x ) := (1 + P xi =1 φ ( i )) · ( φ ( x ) + 1) forevery x ∈ N . • Let h : N → N be the function defined by h (0) := 2 w +6 ( φ ( w ) · f ( η ) · w ) ( η + 3) ,and for every x ∈ N , h ( x + 1) := ( h ( x ) + 2) · w +7 ( φ ( w ) · f ( η ) · w ) ( η + 3) . • Let η := h ( w − . • Let η := η · ( f ( η )) η . • Define λ := η + η .Let G be a graph with no K s,t -subgraph, V = ( V , V , . . . ) a layering of G , L an ( s, V ) -compatible list-assignment, Y a subset of V ( G ) , ( Y , L ) a V -standard pair, and ( T, X ) atree-decomposition of G of V -width at most w such that some bag contains Y , where X =( X t : t ∈ V ( T )) . For every R ⊆ V ( T ) , define X R := S t ∈ R X t . For every U ⊆ V ( G ) , define X | U = ( X t ∩ U : t ∈ V ( T )) . For every integer i with i [ |V| ] , define V i = ∅ .Let r ∗ be the node of T such that X r ∗ contains Y . Consider T to be rooted at r ∗ . Orientthe edges of T away from r ∗ . For each node q of T , define the height of q to be the length of34he path in T from r ∗ to q , and let T q be the subtree of T induced by q and all the descendantsof q . For i ∈ N , define T ( i ) to be the subtree of T induced by the nodes of height at most i .For each vertex v of G , let r v be the node of T closest to r ∗ with v ∈ X r v ; define the height of v to be the height of r v . For i ∈ N , define ∂X V ( T ( i ) ) to be the union of the bags of the nodesof height exactly i .We will construct a desired L -coloring of G by an algorithm. We now give an informaldescription of several notions that are used in the algorithm below. The layers are partitionedinto pairwise disjoint belts, where each belt consists of a very large (but still bounded) set ofconsecutive layers. An interface consists of roughly the last one third of the layers within onebelt, along with the first roughly one third of the layers in the next belt. Then the interior ofan interface is the last few layers of the first belt, along with the first few layers of the nextbelt. The algorithm also uses a linear ordering of V ( G ) that never changes. We associatewith each subgraph of G the first vertex in the ordering that is in the subgraph. We usethis vertex to order subgraphs, whereby a subgraph that has a vertex early in the ordering isconsidered to be “old”. We now formalize these ideas.Define σ T to be a depth-first-search order of T rooted at r ∗ . That is, σ T ( t ) = 0 , and if anode t of T is visited earlier than a node t ′ of T in the depth-first-search starting at r ∗ then σ T ( t ) < σ T ( t ′ ) . For every node t of T , we define i t to be the integer such that σ T ( t ) = i t .Define σ to be a linear order of V ( G ) such that for any distinct vertices u, v , if σ T ( r u ) <σ T ( r v ) , then σ ( u ) < σ ( v ) . For every subgraph H of G , define σ ( H ) := min { σ ( v ) : v ∈ V ( H ) } .Note that for any k ∈ N , any set Z consisting of k consecutive layers, any t ∈ V ( T ) , any V -standard pair ( Y ′ , L ′ ) , any monochromatic component M with respect to any L ′ -coloringin G [ Y ′ ] intersecting Z ∩ X t , there exist at most | Z ∩ X t | kw monochromatic components M ′ with respect to any L ′ -coloring in G [ Y ′ ] intersecting Z ∩ X t such that σ ( M ′ ) < σ ( M ) .A belt of ( T, X ) is a subset of V ( G ) of the form S a + s ∗ i = a +1 V i for some nonnegative integer a with a ≡ (mod s ∗ ). Note that each belt consists of s ∗ layers. So for every belt B of ( T, X ) , ( T, X | B ) is a tree-decomposition of G [ B ] of width at most s ∗ w .For every j ∈ [ |V| − , let I j := ( j + ) s ∗ [ i =( j − ) s ∗ V i and I j := ( j + ) s ∗ +(2 s +5) [ i =( j − ) s ∗ − (2 s +5) V i .I j is called the interface at j . (Note that s ∗ is a multiple of 3, so the indices in the definitionof I j and I j are integers.) For every j ∈ [0 , |V| − , define I j, := js ∗ [ i = js ∗ − ( s +2)+1 V i and I j, := js ∗ [ i = js ∗ − s +2)+1 V i , and define I j, := js ∗ + s +2 [ i = js ∗ +1 V i and I j, := js ∗ +2( s +2) [ i = js ∗ +1 V i . Also define I ◦ j := I j, ∪ I j, and I ◦ j := I j, ∪ I j, . For every j ∈ [ |V| − , define S ◦ j to be the setof all s -segments S intersecting I ◦ j . Note that I ◦ j ⊆ S S ∈S ◦ j S ⊆ I ◦ j ⊆ I j for every j ∈ [ |V| − .35n addition, for every j ∈ [ |V| − , I j is contained in a union of two consecutive belts, so forevery t ∈ V ( T ) , | I j ∩ X t | s ∗ w w .We now give some intuition about the algorithm that follows. The input to the algorithmis a tree-decomposition of G with bounded layered width, a set Y of precolored vertices, anda list assignment L of G . Throughout the algorithm, Y ( i,⋆,⋆ ) refers to the current set of coloredvertices, where the superscript ( i, ⋆, ⋆ ) indicates the stage of the algorithm. This vector isincremented in lexicographic order as the algorithm proceeds. The algorithm starts withstage (0 , − , , during which time we initialize several variables. Throughout the algorithm,a monochromatic component refers to a component of the subgraph of G induced by thecurrent precolored set of vertices (or to be more precise, vertices with one color in their list).The algorithm does a DFS search of the tree T indexing the tree-decomposition, consid-ering the nodes t of T with σ T ( t ) = i in turn (for i = 0 , , , . . . ). The algorithm first buildsa fence around the subgraph of vertices in bags rooted at node t (relative to belts). At stage ( i, − , ⋆ ) , the algorithm tries to isolate the k -th oldest component intersecting some segmentintersecting I ◦ j . Then at stage ( i, , ⋆ ) , the algorithm isolates the other monochromatic com-ponents intersecting X t . In both these stages, ⋆ refers to the color given to vertices aroundthe component that we are trying to isolate. Then in stage ( i, ⋆, ⋆ ) we isolate the fencesassociated with subtree T j,t . Finally, the algorithm moves to the next node in the tree, andbuilds new fences with respect to the next node in the tree.Now we give a more detailed intuition of the algorithm. We will simplify some notationsin this explanation for simplicity. The formal description of the algorithm is described later.During the algorithm, we will construct the following. • Subsets Y ( i,j,k ) of V ( G ) for i ∈ [0 , | V ( T ) | ] , j ∈ {− }∪ [0 , | V ( T ) | +1] and k ∈ [0 , s +2] : Theset Y ( i,j,k ) is the set of precolored vertices at stage ( i, j, k ) , where a vertex is said to be precolored if its list contains only one color. At each stage, we will precolor more vertices,so these sets have the property that Y ( i,j,k ) ⊆ Y ( i ′ ,j ′ ,k ′ ) if ( i, j, k ) is lexicographicallysmaller than ( i ′ , j ′ , k ′ ) . • Subsets F j,t of V ( T ) for j ∈ [ |V| − and t ∈ V ( T ) : Each of these sets F j,t is a union offences. For any fixed j ∈ [ |V| − and node t of T , assuming F ′ j,p is given, where p isthe parent of t , we do the following: – We first construct F j,p as the following: We consider the F ′ j,p ∩ V ( T t ) -parts of T t containing t . Note that it is possible to have more than one such part since t couldbe in F ′ j,p . Then for each such part T ′ , we restrict ourselves to the subgraph of G induced by X V ( T ′ ) ∩ I j . That is, the subgraph induced by the subset of vertices inthe part T ′ and in the closure of the interface at j . Note that this subgraph hasa tree-decomposition naturally given by ( T, X ) . We construct a fence F j,T ′ in thistree-decomposition based on the current set of precoloring vertices Y ( i, − , in thissubgraph, where i = σ ( t ) . Then F j,p is the union of F ′ j,p and F j,T ′ among all suchparts T ′ . – When F j,p is defined, we will extend our precolored set Y ( i, − , to Y ( i +1 , − , bysome procedure that is explained later. After Y ( i +1 , − , is given, we construct F ′ j,t in a similar way as we construct F j,p from F ′ j,p , except we replace F ′ j,p by F j,p andreplace Y ( i, − , by Y ( i +1 , − , . 36he key idea is to make sure that for every F j,p ∩ V ( T t ) -part containing t , the subgraphinduced by this part only contains few precolored vertices. • Subtrees T j,t of T t and the “boundary” ∂T j,t of T j,t , for j ∈ [ |V| − and t ∈ V ( T ) : T j,t isthe union of all F j,p ∩ V ( T t ) -parts containing t , and ∂T j,t is the union of the boundaryof those parts, where t is not included. Roughly speaking, T j,t is a small portion of T t containing t , and X V ( T j,t ) contains only few prepcolored vertices by the construction ofthe fences. • Subsets Z t of X V ( T t ) for every t ∈ V ( T ) : Z t consists of the vertices in X V ( T t ) that areeither not in the interior of any interface, or in the interior of the interface at j for some j but also in X V ( T j,t ) . • Subsets D ( ∗ , ∗ , ∗ ) of V ( G ) : Those sets will help us extend the current set of precoloredvertices. We will explain the usage of these sets later. • Sets E j,t and E ( ∗ , ∗ ) j,t of “fake edges”, for j ∈ [ |V| − and t ∈ V ( T ) : Each fake edge isa pair of two distinct vertices that are contained in the same s -segment in S ◦ j . At anymoment of the algorithm, the current set of precolored vertices form some monochro-matic components, and it is possible that more than one of them will be merged intoa bigger monochromatic component after we further color more vertices. These fakeedges join current monochromatic components that are possibly to be merged into abigger monochromatic component in the future to form “pseudocomponents”. Note thatwhether a pair of two vertices is a fake edge depends on j and t .Now we give a sketch of the algorithm and explain the intuition. Recall that during thealgorithm, Y ( i, ∗ , ∗ ) always contains all vertices of G contained in X t with σ T ( t ) i . So at thevery beginning of the algorithm, we color every uncolored vertex in X r ∗ to get Y (0 , − , . Thenwe proceed the algorithm for i = 0 , , , · · · . Now we fix i to be a nonnegative integer and t the node of T with σ T ( t ) = i , and explain what we will do at this stage. • First, for each j ∈ [ |V| − , we build fences F ′ j,p and F j,p , where p is the parent of t , andsubtrees T j,t of T t . Note that it defines Z t . We will further color vertices in Z t . Theidea for constructing those fences and Z t is to make sure that only a small number ofprecolor vertices in Z t are close to the boundary of two different belts. • Then we move to stage ( i, − , ∗ ) . In this stage, we will “isolate” the monochromaticpseudocomponents in Z t intersecting X t and some segment in S ◦ j for some j ∈ [ |V| − .That is, we will color some vertices in Z t to stop the growth of those monochromaticcomponents by coloring their gates by using a color that is different from the colorof the monochromatic pseudocomponent. Note that we cannot isolate all such compo-nents at once, since some vertex can be a gate of two monochromatic pseudocomponentswith different colors. The way we break the tie is according to the ordering σ of thosemonochromatic components. For each j ∈ [ |V| − , we will isolate the monochromaticpseudocomponent with minimum σ -order, and then isolate the monochromatic pseu-docomponent with second minimum σ -order and so on. Say we are isolating the k -thmonochromatic pseudocomponent among all those monochromatic pseudocomponentsintersecting X t and some segment in S ◦ j for each j . Then W ( i, − ,k )0 consists of the ver-tices in the k -th monochromatic pseudocomponent for any j . Then for each j , thereuniquely exists a color q + 1 such that this monochromatic pseudocomponent uses color37 + 1 . We write W ( i, − ,k,q +1)1 to denote the vertices in W ( i, − ,k )0 with color q + 1 . Thenwe color the gates of W ( i, − ,k,q +1)1 contained in Z t by using a color different from q + 1 .Then we move to color the ( k + 1) -th monochromatic pseudocomponent, and so one.Note that the ( k + 1) -th monochromatic pseudocomponent can become larger due tothe above process of coloring gates. • Now we move to stage ( i, , ∗ ) . In this stage we isolated all monochromatic(pseudo)components in Z t intersecting X t . Note that all monochromatic pseudocom-ponents intersecting some s -segment in S ◦ j for some j ∈ [ |V| − are isolated in theprevious stage. So all the remaining monochromatic pseudocomponents are disjointfrom the s -segments in S ◦ j for every j ∈ [ |V| − , and hence they are disjoint fromall fake edges. Hence all monochromatic pseudocomponents that are dealt with at thisstage are indeed monochromatic components. We will isolate all such monochromaticcomponents with color 1, and then the ones with color 2 and so on. • Now we move to stage ( i, > , ∗ ) . In this stage we restrict to X V ( T j,t ) ∩ I j , for each j ∈ [ |V| − . We denote the union of them by W ( i )4 . Now we fix j to be an integer in [ |V| − . The objective is that at the end of this stage, the monochromatic componentscontained in X V ( T j,t ) ∩ I j intersecting ∂T j,t will not have more new vertices in X V ( T j,t ) ∩ I j in the future. The idea for achieving this is to color X t ′ ∩ I j for some nodes t ′ ∈ ∂T j,t in the following way. Consider a set of “base nodes”. At the beginning, the only basenode is t . If there exists a monochromatic path P from a base node to a node t ′ in ∂T j,t , then we add t ′ into the set of base nodes and color X t ′ ∩ I j , unless the vertexin V ( P ) ∩ X t ′ belongs to D ( i, ∗ , ∗ ) − X t or in the intersection of D ( i, ∗ , ∗ ) ∩ X t and a bagof a some base nodes at stage ( i ′ , ∗ , ∗ ) for some i ′ < i , where D ( i, ∗ , ∗ ) is a special set ofvertices roughly consists of all vertices contained in the previous monochromatic pathsfor generating new base nodes. Then we isolate those newly colored vertices by coloringtheir gates. It might create new monochromatic paths from base nodes to other node in ∂T j,t . We repeat the above process until no new base nodes can be added. The intuitionof considering D ( i, ∗ , ∗ ) is to make sure that we only include “necessary” nodes in ∂T j,t inthe set of base nodes in order to limit the size of the set of precolored vertices. • Then we move to the stage for adding new fake edges. Recall that when we colormore vertices, we always try to isolate some monochromatic pseudocomponents. Butsuch isolation is “incomplete” because of the definition of Z t . In addition, the isolationprocess depends on the order of the pseudocomponents. So we more or less require someextra information to indicate whether two monochromatic pseudocomponents will bejoined into one in the future so that they should be treated the same when we order thepseudocomponents. This is the purpose of the fake edges. Whether a pair of verticesform a fake edge depends on where we “observe” it. That is, when we “stand on” a node t ′ of T , we consider whether we should add a fake edge to connect two vertices if in thefuture there will be monochormatic paths in X V ( T t ) connecting these two vertices. So itis only relavant when t ′ ∈ V ( T t ) .We fix a node t ′ ∈ V ( T t ) . Each fake edge is a pair of distinct vertices u, v in X t ′′ ∩ S forsome j ∈ [ |V| − , t ′′ ∈ ∂T j,t and S ∈ S ◦ j . Also, u and v must be already colored by thesame color. So there exist monochromatic pseudocomponents M u and M v containing u and v , respectively. And M u = M v , otherwise u and v are already contained in38he same monochromatic pseudocomponent and there is no need to add a fake edgeto connected u and v . By symmetry, we may assume σ ( M u ) < σ ( M v ) . Note thatwe want to add a fake edge to link u and v if it is possible that M u and M v will bejoined into one monochormatic component in the future. So each M u and M v musthave gates contained in X V ( T t ′′ ) − X t ′′ . Recall that during the isolation process, themonochormatic psuedocomponent that with minimum σ -order will be isolated the first,so that it will not grow in the future. Hence M u and M v cannot be the monochromaticpseudocomponent with the minimum σ -order.In addition, the need of adding a fake edge is “triggered” by another monochro-matic pseudocomponent intersecting X t with smaller σ -value. That is, there existsa monochromatic pseudocomponent intersecting X t and X t ′′ with smaller σ -order than σ ( M u ) and σ ( M v ) so that isolating this monochromatic pseudocomponent might leadto possble needs for coloring more vertices to eventually make M u and M v be mergedinto one monochromatic pseudocomponent. Hence, there is no danger for merging M u and M v into one monochromatic pseudocomponent in the future if the gates of thosemonochromatic pseudocomponents with smaller σ -value than σ ( M u ) and σ ( M v ) areseparated by a “think buffer” from the gates of M u and M v . Here a “thick buffer” meansa sequence of nodes that is similar to a fan defined in the previous section, so it is asequence of many nodes of T such that they are more or less have disjoint bags, andthere exist a directed path in T passing through all of them. • Finally, color all vertices in the bag of the node with σ T -value i + 1 , build fences F ′ j,t and move to the next i . Stage (0 , − , : Initialization • Let ( Y (0 , − , , L (0 , − , ) be an ( X r ∗ , -progress of ( Y , L ) . • Let U (0 , − := ∅ and D (0 , , := ∅ . • Let F j, := ∅ for every j ∈ [ |V| − . • Let S (0 , j,t := ∅ , S (0 , j,t := ∅ and S (0 , j,t := ∅ for every j ∈ [ | V | − and t ∈ V ( T ) . • Let E (0) j,t := ∅ for every j ∈ [ |V| − and t ∈ V ( T ) .For i = 0 , , , . . . , let t be the node of T with σ T ( t ) = i and define the following: Stage: Building Fences • Define the following: – For every j ∈ [ |V| − , ∗ Let F ′ j,p := F j, if t = r ∗ , and let p be the parent of t if t = r ∗ , ∗ For every F ′ j,p ∩ V ( T t ) -part T ′ of T t containing t , define F j,T ′ to be a ( T ′ , X | X V ( T ′ ) ∩ I j , ( Y ( i, − , ∪ X ∂T ′ ) ∩ X V ( T ′ ) ∩ I j ) -fence. ∗ Let F j,p := F ′ j,p ∪ S T ′ F j,T ′ , where the union is over all F ′ j,p ∩ V ( T t ) -parts T ′ of T t containing t . ∗ Let T j,t := S T ′ T ′ , where the union is over all F j,p ∩ V ( T t ) -parts T ′ of T t containing t . 39 Let ∂T j,t := S T ′ ( ∂T ′ − { t } ) , where the union is over all F j,p ∩ V ( T t ) -parts T ′ of T t containing t . – Let Z t := ( X V ( T t ) − S |V|− j =1 I ◦ j ) ∪ ( S |V|− j =1 ( I ◦ j ∩ X V ( T j,t ) )) . • Let U ( i, − , := U ( i, − . Stage ( i, − , ⋆ ) : Isolate the k -th Oldest Component Intersecting a Segment in S ◦ j : • For each k ∈ [0 , w − , define the following: – For every j ∈ [ |V| − , define M ( t ) j,k to be the monochromatic E ( i ) j,t -pseudocomponentin G [ Y ( i, − ,k ) ] such that σ ( M ( t ) j,k ) is the smallest among all monochromatic E ( t ) j,t -pseudocomponents M in G [ Y ( i, − ,k ) ] intersecting X t with A L ( i, − ,k ) ( V ( M )) ∩ X V ( T t ) − X t = ∅ and contained in some s -segment S ∈ S ◦ j whose level equals the color of M such that V ( M ) ∩ S k − ℓ =0 V ( M ( t ) j,ℓ ) = ∅ . – Let W ( i, − ,k )0 := X V ( T t ) ∩ S j ∈ [ |V|− V ( M ( t ) j,k ) . – Let ( Y ( i, − ,k, , L ( i, − ,k, ) = ( Y ( i, − ,k ) , L ( i, − ,k ) ) . – For every q ∈ [0 , s + 1] , define the following: ∗ Let W ( i, − ,k,q )1 := { v ∈ W ( i, − ,k )0 : L ( i, − ,k ) ( v ) = q + 1 } . ∗ Let W ( i, − ,k,q )2 := A L ( i, − ,k,q ) ( W ( i, − ,k,q )1 ) ∩ Z t . ∗ Let ( Y ( i, − ,k,q +1) , L ( i, − ,k,q +1) ) be a ( W ( i, − ,k,q )2 , q + 1) -progress of ( Y ( i, − ,k,q ) , L ( i, − ,k,q ) ) . ∗ If ( Y ( i, − ,k,q +1) − Y ( i, − ,k,q ) ) ∩ I j ∩ Z t = ∅ and c ( t ) j is undefined, then let c ( t ) j := q + 1 . – Let ( Y ( i, − ,k +1) , L ( i, − ,k +1) ) := ( Y ( i, − ,k,s +2) , L ( i, − ,k,s +2) ) . – Let U ( i, − ,k +1) := U ( i, − ,k ) ∪ W ( i, − ,k )0 . Stage ( i, , ⋆ ) : Isolate the other components intersecting X t • Let ( Y ( i, , , L ( i, , ) := ( Y ( i, − ,w ) , L ( i, − ,w ) ) . • Let U ( i, := U ( i, − ,w ) . • For each k ∈ [0 , s + 1] , define the following: – Let W ( i, ,k )0 := { v ∈ Y ( i, ,k ) ∩ X V ( T t ) : L ( i, ,k ) ( v ) = { k + 1 } andthere exists a path P in G [ Y ( i, ,k ) ∩ X V ( T t ) ] from v to X t such that L ( i, ,k ) ( q ) = { k + 1 } for all q ∈ V ( P ) } . – Let W ( i, ,k )2 := A L ( i, ,k ) ( W ( i, ,k )0 ) ∩ Z t . – Let ( Y ( i, ,k +1) , L ( i, ,k +1) ) be a ( W ( i, ,k )2 , k + 1) -progress of ( Y ( i, ,k ) , L ( i, ,k ) ) . – Let U ( i,k +1) := U ( i,k ) ∪ W ( i, ,k )0 . Stage ( i, > , ⋆ ) : Isolating the Fences of T j,t • Let W ( i, − := S |V|− j =1 ( X t ∩ I j ) . • Let W ( i )4 := S |V|− j =1 ( X V ( T j,t ) ∩ I j ) . • For each ℓ ∈ [0 , | V ( T ) | ] , define the following:40 Let W ( i,ℓ )3 := W ( i,ℓ − ∪ S |V|− j =1 S q ( X q ∩ I j ) , where the last union is over all nodes q in ∂T j,t for which there exists a monochromatic path in G [ Y ( i,ℓ,s +2) ∩ I j ∩ W ( i )4 ] from W ( i,ℓ − to X q ∩ I j − (( D ( i,ℓ, − X t ) ∪ { v ∈ D ( i,ℓ, ∩ X t ∩ X q ′ ∩ I j : q ′ ∈ V ( T t ) − { t } ,q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 for some i ′ ∈ [0 , i − and ℓ ′ ∈ [0 , | V ( T ) | ] } ) internally disjoint from X t ∪ X ∂T j,t . – Let ( Y ( i,ℓ +1 , , L ( i,ℓ +1 , ) be a ( W ( i,ℓ )3 , -progress of ( Y ( i,ℓ,s +2) , L ( i,ℓ,s +2) ) . – For each k ∈ [0 , s + 1] , define the following: ∗ Let W ( i,ℓ +1 ,k )0 := { v ∈ Y ( i,ℓ +1 ,k ) ∩ W ( i )4 : L ( i,ℓ +1 ,k ) ( v ) = { k + 1 } and there exists a path P in G [ Y ( i,ℓ +1 ,k ) ∩ W ( i )4 ] from v to W ( i,ℓ )3 such that L ( i,ℓ +1 ,k ) ( q ) = { k + 1 } for all q ∈ V ( P ) } . ∗ Let W ( i,ℓ +1 ,k )2 := A L ( i,ℓ +1 ,k ) ( W ( i,ℓ +1 ,k )0 ) ∩ W ( i )4 . ∗ Let ( Y ( i,ℓ +1 ,k +1) , L ( i,ℓ +1 ,k +1) ) be the ( W ( i,ℓ +1 ,k )2 , k + 1) -progressof ( Y ( i,ℓ +1 ,k ) , L ( i,ℓ +1 ,k ) ) . ∗ Let D ( i,ℓ,k +1) := D ( i,ℓ,k ) ∪ W ( i,ℓ +1 ,k )0 . – Let D ( i,ℓ +1 , := D ( i,ℓ,s +2) . Stage: Adding Fake Edges • For each j ∈ [ |V| − and t ′ ∈ V ( T ) − ( V ( T t ) − { t } ) , let E ( i +1) j,t ′ := E ( i ) j,t ′ . • For each j ∈ [ |V| − , let E ( i, j,t := E ( i ) j,t . • For each j ∈ [ |V| − and t ′ ∈ V ( T t ) − { t } , let E ( i +1) j,t ′ := E ( i, | V ( G ) | ) j,t , where every ℓ ∈ [0 , | V ( G ) | − , let E ( i,ℓ +1) j,t be the union of E ( i,ℓ ) j,t and the set consisting of the2-element sets { u, v } satisfying the following. – { u, v } ⊆ Y ( i, | V ( T ) | +1 ,s +2) . – L ( i, | V ( T ) | +1 ,s +2) ( u ) = L ( i, | V ( T ) | +1 ,s +2) ( v ) . – There exists an s -segment S in S ◦ j whose level equals the color of u and v suchthat { u, v } ⊆ S . – There exists t ′′ ∈ ∂T j,t such that ∗ { u, v } ⊆ X t ′′ , ∗ V ( M ) ∩ X t ′′ = ∅ , where M is the monochromatic E ( i,ℓ ) j,t -pseudocomponentin G [ Y ( i, | V ( T ) | +1 ,s +2) ] such that σ ( M ) is the ( ℓ + 1) -th smallest among allmonochromatic E ( i,ℓ ) j,t -pseudocomponents in G [ Y ( i, | V ( T ) | +1 ,s +2) ] , ∗ M is contained in some s -segment in S ◦ j whose level equals the color of M , ∗ t ′′ is a witness for X t ′′ ∩ I j ⊆ W ( i,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , ∗ A L ( i, | V ( T ) | +1 ,s +2) ( V ( M )) ∩ X V ( T t ′′ ) − X t ′′ = ∅ , ∗ A L ( i, | V ( T ) | +1 ,s +2) ( V ( M u )) ∩ X V ( T t ′′ ) − X t ′′ = ∅ , ∗ A L ( i, | V ( T ) | +1 ,s +2) ( V ( M v )) ∩ X V ( T t ′′ ) − X t ′′ = ∅ , and ∗ σ ( M ) < min { σ ( M u ) , σ ( M v ) } , 41here M u and M v are the monochromatic E ( i,ℓ ) j,t -pseudocomponents in G [ Y ( i, | V ( T ) | +1 ,s +2) ] containing u and v , respectively. – If ( V ( M u ) ∪ V ( M v )) ∩ X t = ∅ , then at least one of the following does not hold. ∗ for every monochromatic E ( i,ℓ ) j,t -pseudocomponent M ′ in G [ Y ( i, | V ( T ) | +1 ,s +2) ] with σ ( M ′ ) σ ( M ) such that M ′ is contained in some s -segment in S ◦ j whose levelequals the color of M ′ , M ′ satisfies that A L ( i, | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ) − X t ⊆ X V ( T t ′′ ) − ( X t ′′ ∪ Z t ) , ∗ there exists a parade ( t − β u,v,ℓ , t β u,v,ℓ +1 , . . . , t − , t , t , . . . , t α u,v,ℓ ) for some α u,v,ℓ ∈ N and β u,v,ℓ ∈ N such that · t ∈ V ( T t ′′ ) − { t ′′ } , · α u,v,ℓ = ψ ( α u,ℓ , α v,ℓ ) , β u,v,ℓ = ψ ( α u,ℓ , α v,ℓ ) ,where α u,ℓ and α v,ℓ are the integers such that σ ( M u,ℓ ) and σ ( M v,ℓ ) arethe α u,ℓ -th and the α v,ℓ -th smallest among all monochromatic E ( i,ℓ ) j,t -pseudocomponents M ′ in G [ Y ( i, | V ( T ) | +1 ,s +2) ] contained in some s -segment in S ◦ j whose color equals the color of M u and M v with A L ( i, | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ′′ ) − X t ′′ = ∅ , respectively, and M , M , . . . , M β (for some β ∈ N ) are the monochromatic E ( i,ℓ ) j,t -pseudocomponents M ′′ intersecting X t ′′ with A L ( i, | V ( T ) | +1 ,s +2) ( V ( M ′′ )) ∩ X V ( T t ′′ ) − X t ′′ = ∅ contained in some s -segment in S ◦ j whose color equalsthe level of M ′′ such that σ ( M ) < σ ( M ) < · · · < σ ( M β ) and V ( M γ ) = ∅ for every γ > β + 1 , · if β u,v,ℓ = 0 , then t − β u,v,ℓ ∈ V ( T t ′′ ) − { t ′′ } , · for any distinct ℓ , ℓ ∈ [ − β u,v,ℓ , α u,v,ℓ ] − { } , X t ℓ ∩ I j − X t ′′ and X t ℓ ∩ I j − X t ′′ are disjoint non-empty sets, · S M ′′ ( A L ( i, | V ( T ) | +1 ,s +2) ( V ( M ′′ )) ∩ X V ( T t ) − X t ) ⊆ ( X V ( T tαu,v,ℓ ) − X t αu,v,ℓ ) ∩ I ◦ j ,where the union is over all monochromatic E ( i,ℓ ) j,t -pseudocomponents M ′′ in G [ Y ( i, | V ( T ) | +1 ,s +2) ] such that V ( M ′′ ) is contained in some s -segment in S ◦ j whose level equals the color of M ′′ , σ ( M ′′ ) σ ( M ) , V ( M ′′ ) ∩ X t ′′ = ∅ ,and A L ( i, | V ( T ) | +1 ,s +2) ( V ( M ′′ )) ∩ X V ( T t ′′ ) − X t ′′ = ∅ , · there exists x ℓ ∈ { u, v } such that A L ( i, | V ( T ) | +1 ,s +2) ( V ( M x ℓ )) ∩ ( X V ( T t ) − X t ) ∩ X V ( T t ) − X t = ∅ , and if β u,v,ℓ = 0 , then A L ( i, | V ( T ) | +1 ,s +2) ( V ( M x ℓ )) ∩ ( X V ( T t ) − X t ) ∩ X V ( T q − βu,v,ℓ ) − ( X q − βu,v,ℓ ∪ I ◦ j ) = ∅∗ there exists a monochromatic E ( i ) j,t -pseudocomponent M ∗ in G [ Y ( i, | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) σ ( M ) such that M ∗ is contained in some s -segment in S ◦ j whoselevel equals the color of M ∗ satisfying that A L ( i, | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X V ( T t ) − X t ⊆ X V ( T t ′′ ) − ( X t ′′ ∪ Z t ) . – There do not exist q ∈ V ( T ) with T j,t ⊆ T j,q and i q < i t , a witness q ′ ∈ ∂T j,q for X q ′ ∩ I j ⊆ W ( i q ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i q ) j,q -pseudocomponent in G [ Y ( i q , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and { u, v } . Stage: Moving to the Next Node in the Tree Let ( Y ( i +1 , − , , L ( i +1 , − , ) be a ( X t ′ , -progress of ( Y ( i, | V ( T ) | +1 ,s +2) , L ( i, | V ( T ) | +1 ,s +2) ) ,where t ′ is the node of T with σ T ( t ′ ) = i + 1 . • Let U ( i +1 , − := U ( i,s +2) . • Let D ( i +1 , , := D ( i, | V ( T ) | ,s +2) . • For every node t ′ with t ′ ∈ V ( T t ) − { t } and j ∈ [ | V | − , – If t ′ ∈ V ( T j,t ) − ∂T j,t , then define ∗ S ( i +1 , j,t ′ := S ( i, j,t ′ ∪ S | V ( T ) | ℓ =0 S s +1 k =0 W ( i,ℓ +1 ,k )0 , and ∗ S ( i +1 , j,t ′ := S ( i, j,t ′ ∪ S |V|− j =1 S q ( X q ∩ I j ) , where the last union is over all nodes q of T such that q ∈ V ( T t ′ ) − { t ′ } and q is a witness for X q ∩ I j ⊆ W ( i,ℓ ′ )3 forsome ℓ ′ ∈ [0 , | V ( T ) | ] ; ∗ S ( i +1 , j,t ′ := S ( i, j,t ′ ∪ S w − k =0 W ( i, − ,k )0 ∪ S s +1 k =0 W ( i, ,k )0 . – otherwise, define S ( i +1 , j,t ′ := S ( i, j,t ′ , S ( i +1 , j,t ′ := S ( i, j,t ′ and S ( i +1 , j,t ′ := S ( i, j,t ′ . Stage: Building a New Fence • For every j ∈ [ |V| − , define the following: – For every F j,p ∩ V ( T t ) -part T ′ of T t containing t , let F ′ j,T ′ be a ( T ′ , X | X V ( T ′ ) ∩ I j , ( Y ( i +1 , − , ∪ X ∂T ′ ) ∩ X V ( T ′ ) ∩ I j ) -fence. – Let F ′ j,t := F j,p ∪ S T ′ F ′ j,T ′ , where the union is over all F j,p ∩ V ( T t ) -parts T ′ of T t containing t .It is clear that if ( i ′ , j ′ , k ′ ) is a triple lexicographically smaller than a triple ( i, j, k ) , then L ( i,j,k ) ( v ) ⊆ L ( i ′ ,j ′ ,k ′ ) ( v ) for every v ∈ V ( G ) , so Y ( i,j,k ) ⊇ Y ( i ′ ,j ′ ,k ′ ) . In addition, X t ⊆ Y ( i t , − , .Recall that i t is the number such that σ T ( t ) = i t . In particular, we have | L ( | V ( T ) | , − , ( v ) | = 1 for every v ∈ V ( G ) , so there exists a unique L ( | V ( T ) | , − , -coloring c of G . By construction, c is an L -coloring. We prove below that c has clustering λ . Claim 4.12.1. Let i ∈ N , and let t ∈ V ( T ) with i t = i . Let j ∈ [ |V|− and ℓ ∈ [ − , | V ( T ) | ] .If k ∈ [0 , s +1] and P is a c -monochromatic path of color k +1 contained in G [ W ( i )4 ] intersecting W ( i,ℓ )3 , then V ( P ) ⊆ Y ( i,ℓ +1 ,k ) and A L ( i,ℓ +1 ,k +1) ( V ( P )) ∩ W ( i )4 = ∅ .Proof. First suppose that V ( P ) Y ( i,ℓ +1 ,k ) . Since W ( i,ℓ )3 ⊆ Y ( i,ℓ +1 , , V ( P ) ∩ W ( i,ℓ )3 isa nonempty subset of Y ( i,ℓ +1 ,k ) . So there exists a longest subpath P ′ of P contained in G [ Y ( i,ℓ +1 ,k ) ] intersecting W ( i,ℓ )3 . Since V ( P ′ ) ⊆ V ( P ) ⊆ W ( i )4 , V ( P ′ ) ⊆ W ( i,ℓ +1 ,k )0 . In ad-dition, P = P ′ , for otherwise V ( P ) = V ( P ′ ) ⊆ Y ( i,ℓ +1 ,k ) , a contradiction. So there ex-ist v ∈ V ( P ′ ) and u ∈ N P ( v ) − V ( P ′ ) . That is, c ( u ) = k + 1 and u Y ( i,ℓ +1 ,k ) . So u ∈ A L ( i,ℓ +1 ,k ) ( { v } ) ∩ V ( P ) − W ( i,ℓ +1 ,k )0 ⊆ A L ( i,ℓ +1 ,k ) ( W ( i,ℓ +1 ,k )0 ) ∩ W ( i )4 ⊆ W ( i,ℓ +1 ,k )2 . But ( Y ( i,ℓ +1 ,k +1) , L ( i,ℓ +1 ,k +1) ) is a ( W ( i,ℓ +1 ,k )2 , k + 1) -progress, so k + 1 L ( i,ℓ +1 ,k +1) ( u ) . Hence c ( u ) = k + 1 , a contradiction.Now we suppose that A L ( i,ℓ +1 ,k +1) ( V ( P )) ∩ W ( i )4 = ∅ . So there exists z ∈ A L ( i,ℓ +1 ,k +1) ( V ( P )) ∩ W ( i )4 . Note that we have shown that V ( P ) ⊆ Y ( i,ℓ +1 ,k ) , so V ( P ) ⊆ W ( i,ℓ +1 ,k )0 and A L ( i,ℓ +1 ,k +1) ( V ( P )) ⊆ A L ( i,ℓ +1 ,k ) ( V ( P )) . In addition, z ∈ A L ( i,ℓ +1 ,k +1) ( V ( P )) , so z Y ( i,ℓ +1 ,k +1) ⊇ Y ( i,ℓ +1 ,k ) ⊇ W ( i,ℓ +1 ,k )0 . So z ∈ A L ( i,ℓ +1 ,k +1) ( V ( P )) ∩ W ( i )4 − W ( i,ℓ +1 ,k )0 ⊆ A L ( i,ℓ +1 ,k ) ( V ( P )) ∩ W ( i )4 − W ( i,ℓ +1 ,k )0 ⊆ W ( i,ℓ +1 ,k )2 . Since ( Y ( i,ℓ +1 ,k +1) , L ( i,ℓ +1 ,k +1) ) is a43 W ( i,ℓ +1 ,k )2 , k + 1) -progress of ( Y ( i,ℓ +1 ,k ) , L ( i,ℓ +1 ,k ) ) , we know k + 1 L ( i,ℓ +1 ,k +1) ( z ) , so z A L ( i,ℓ +1 ,k +1) ( V ( P )) , a contradiction. This proves the claim. Claim 4.12.2. Let i, i ′ ∈ N with i ′ < i , and let t ∈ V ( T ) with i t = i . Let j ∈ [ | V | − and ℓ ∈ [0 , | V ( T ) | ] . If Y ( i ′ ,ℓ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t , then ( W ( i ′ ,ℓ )3 − W ( i ′ ,ℓ − ) ∩ I j = ∅ , and either • ℓ > , or • | X t ∩ I j ∩ D ( i ′ ,ℓ,s +2) | > | X t ∩ I j ∩ D ( i ′ ,ℓ, | , or • | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | , or • | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | .Proof. Let t ′ be the node of T with i t ′ = i ′ . Since Y ( i ′ ,ℓ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t , t ′ is an ancestor of t .Suppose that ( W ( i ′ ,ℓ )3 − W ( i ′ ,ℓ − ) ∩ I j = ∅ . Then for every k ∈ [0 , s + 1] and everymonochromatic path P of color k + 1 in G [ Y ( i ′ ,ℓ +1 ,k ) ∩ W ( i ′ )4 ] intersecting W ( i ′ ,ℓ )3 ∩ I j , P is a c -monochromatic path in G [ W ( i ′ )4 ] intersecting W ( i ′ ,ℓ )3 ∩ I j ⊆ W ( i ′ ,ℓ − ∩ I j , so Claim 4.12.1 impliesthat V ( P ) ⊆ Y ( i ′ ,ℓ,k ) ∩ W ( i ′ )4 , and hence V ( P ) ⊆ W ( i ′ ,ℓ,k )0 . Hence W ( i ′ ,ℓ +1 ,k )0 ∩ I j ⊆ W ( i ′ ,ℓ,k )0 ∩ I j for every k ∈ [0 , s + 1] . Since no edge of G is between I j and I j ′ for any j ′ = j , for every k ∈ [0 , s + 1] , W ( i ′ ,ℓ +1 ,k )2 ∩ I j = A L ( i ′ ,ℓ +1 ,k ) ( W ( i ′ ,ℓ +1 ,k )0 ) ∩ W ( i ′ )4 ∩ I j = A L ( i ′ ,ℓ +1 ,k ) ( W ( i ′ ,ℓ +1 ,k )0 ∩ I j ) ∩ W ( i ′ )4 ∩ I j ⊆ A L ( i ′ ,ℓ +1 ,k ) ( W ( i ′ ,ℓ,k )0 ∩ I j ) ∩ W ( i ′ )4 ∩ I j ⊆ A L ( i ′ ,ℓ +1 ,k ) ( W ( i ′ ,ℓ,k )0 ) ∩ W ( i ′ )4 ∩ I j ⊆ A L ( i ′ ,ℓ,k ) ( W ( i ′ ,ℓ,k )0 ) ∩ W ( i ′ )4 ∩ I j = W ( i ′ ,ℓ,k )2 ∩ I j Hence ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ I j ⊆ S s +1 k =0 ( W ( i ′ ,ℓ +1 ,k )2 ∩ I j ) ⊆ S s +1 k =0 ( W ( i ′ ,ℓ,k )2 ∩ I j ) ⊆ Y ( i ′ ,ℓ,s +2) ∩ I j . So Y ( i ′ ,ℓ +1 ,s +2) ∩ I j ⊆ Y ( i ′ ,ℓ,s +2) ∩ I j , a contradiction.Hence ( W ( i ′ ,ℓ )3 − W ( i ′ ,ℓ − ) ∩ I j = ∅ .Suppose to the contrary that ℓ = 0 , | X t ∩ I j ∩ D ( i ′ ,ℓ,s +2) | | X t ∩ I j ∩ D ( i ′ ,ℓ, | , | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | , and | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | . Since D ( i ′ ,ℓ,s +2) ⊇ D ( i ′ ,ℓ, , S ( i ′ +1 , j,t ⊇ S ( i ′ , j,t and S ( i ′ +1 , j,t ⊇ S ( i ′ , j,t , we know X t ∩ I j ∩ D ( i ′ ,ℓ,s +2) = X t ∩ I j ∩ D ( i ′ ,ℓ, , X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t and X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t .Since ( W ( i ′ ,ℓ )3 − W ( i ′ ,ℓ − ) ∩ I j = ∅ , there exists q ∈ ∂T j,t ′ such that q is a witness for X q ∩ I j ⊆ W ( i ′ ,ℓ )3 and X q ∩ I j W ( i ′ ,ℓ − . So X q ∩ I j is a nonempty subset of W ( i ′ ,ℓ )3 ∩ I j with X q ∩ I j W ( i ′ ,ℓ − , and there exists a c -monochromatic path P in G [ Y ( i ′ ,ℓ,s +2) ∩ I j ∩ W ( i ′ )4 ] from W ( i ′ ,ℓ − to X q ∩ I j − (( D ( i ′ ,ℓ, − X t ′ ) ∪ { v ∈ D ( i ′ ,ℓ, ∩ X t ′ ∩ X q ′ ∩ I j : q ′ ∈ V ( T t ′ ) − { t ′ } , q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′′ ,ℓ ′ )3 for some i ′′ ∈ [0 , i ′ − and ℓ ′ ∈ [0 , | V ( T ) | ] } ) and internallydisjoint from X t ′ ∪ X ∂T j,t ′ . Furthermore, choose such a node q such that q ∈ V ( T t ) − { t } ifpossible. Note that there exists k ∈ [0 , s + 1] such that the color of P is k + 1 .44e first suppose that q ∈ V ( T t ) − { t } . Since V ( P ) ⊆ Y ( i ′ ,ℓ,s +2) ∩ W ( i ′ )4 ⊆ Y ( i ′ ,ℓ +1 ,k ) ∩ W ( i ′ )4 , V ( P ) ⊆ W ( i ′ ,ℓ +1 ,k )0 ⊆ D ( i ′ ,ℓ,k +1) ⊆ D ( i ′ ,ℓ,s +2) . Since X t ∩ I j ∩ D ( i ′ ,ℓ,s +2) = X t ∩ I j ∩ D ( i ′ ,ℓ, , V ( P ) ∩ X t ⊆ D ( i ′ ,ℓ, . Let z be the end of P belonging to X q ∩ I j − (( D ( i ′ ,ℓ, − X t ′ ) ∪ { v ∈ D ( i ′ ,ℓ, ∩ X t ′ ∩ X q ′ ∩ I j : q ′ ∈ V ( T t ′ ) − { t ′ } , q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′′ ,ℓ ′ )3 for some i ′′ ∈ [0 , i ′ − and ℓ ′ ∈ [0 , | V ( T ) | ] } ) , such that z W ( i ′ ,ℓ − if possible.Suppose z D ( i ′ ,ℓ, . Since X t ∩ V ( P ) ⊆ D ( i ′ ,ℓ, , z X t . Since ℓ = 0 , P has one endin W ( i ′ , − . Since t ′ is an ancestor of t and q ∈ V ( T t ) , V ( P ) intersects X t , and there existsa subpath P ′′ of P from X t ∩ V ( P ) ⊆ D ( i ′ ,ℓ, to z ∈ X V ( T t ) − ( X t ∪ D ( i ′ ,ℓ, ) and internallydisjoint from X t . So there exist a ∈ V ( P ′′ ) ∩ D ( i ′ ,ℓ, and b ∈ N P ′′ ( a ) − D ( i ′ ,ℓ, . Note that b ∈ V ( P ′′ ) − D ( i ′ ,ℓ, ⊆ V ( P ) − D ( i ′ ,ℓ, . Since X t ∩ V ( P ) ⊆ D ( i ′ ,ℓ, , b ∈ N P ′′ ( a ) − ( D ( i ′ ,ℓ, ∪ X t ) .Let I ∗ = { i ′′ ∈ [0 , i ′ − 1] : a ∈ S | V ( T ) | ℓ ′ =0 W ( i ′′ ,ℓ ′ +1 ,k )0 } . Since ℓ = 0 and a ∈ D ( i ′ ,ℓ, , a ∈ D ( i ′ , , = D ( i ′ − , | V ( T ) | ,s +2) . Since c ( a ) = k + 1 , a ∈ S i ′ − i ′′ =0 S | V ( T ) | ℓ ′ =0 W ( i ′′ ,ℓ ′ +1 ,k )0 , so I ∗ = ∅ . Note that forevery i ′′ ∈ I ∗ , there exists ℓ i ′′ ∈ [0 , | V ( T ) | ] such that either ℓ i ′′ = 0 and a ∈ W ( i ′′ ,ℓ i ′′ +1 ,k )0 ,or ℓ i ′′ > and a ∈ W ( i ′′ ,ℓ i ′′ +1 ,k )0 − W ( i ′′ ,ℓ i ′′ ,k )0 , so there exists a c -monochromatic path P i ′′ ofcolor k + 1 in G [ Y ( i ′′ ,ℓ i ′′ +1 ,k ) ∩ W ( i ′′ )4 ] from a to W ( i ′′ ,ℓ i ′′ )3 and internally disjoint from W ( i ′′ ,ℓ i ′′ )3 .Since b D ( i ′ ,ℓ, , b Y ( i ′′ ,ℓ i ′′ +1 ,k ) ∩ W ( i ′′ )4 for every i ′′ ∈ I ∗ . For every i ′′ ∈ I ∗ , since A L ( i ′′ ,ℓi ′′ +1 ,k +1) ( V ( P i ′′ )) ∩ W ( i ′′ )4 = ∅ by Claim 4.12.1, we have b W ( i ′′ )4 (since b ∈ W ( i ′′ )4 impliesthat b Y ( i ′′ ,ℓ i ′′ +1 ,k ) ⊇ V ( P i ′′ ) and hence b ∈ A L ( i ′′ , ,k +1) ( V ( P i ′′ )) ∩ W ( i ′′ )4 , a contradiction).Since P ′′ is from X t to X q and internally disjoint from X t ∪ X t ′ ∪ X ∂T j,t ′ , and since b W ( i ′′ )4 ,we know for every i ′′ ∈ I ∗ , q V ( T j,t i ′′ ) for any node t i ′′ of T of height i ′′ . Since q ∈ ∂T j,t ′ , forevery i ′′ ∈ I ∗ , if some node in ∂T j,t i ′′ belongs to the path in T from t ′ to the parent of q , thenthis node must be t ′ . This together with the fact q V ( T j,t i ′′ ) for every i ′′ ∈ I ∗ and every node t i ′′ of T of height i ′′ , we have t V ( T j,t i ′′ ) for every i ′′ ∈ I ∗ and every node t i ′′ of T of height i ′′ . If a ∈ S ( i ′′ +1 , j,t − S ( i ′′ , j,t for some i ′′ ∈ [0 , i ′ − , then a ∈ S | V ( T ) | ℓ ′ =0 W ( i ′′ ,ℓ ′ +1 ,k )0 , so i ′′ ∈ I ∗ , butsince a ∈ S | V ( T ) | ℓ ′ =0 W ( i ′′ ,ℓ ′ +1 ,k )0 ⊆ W ( i ′′ )4 implies that t ∈ V ( T j,t i ′′ ) for some node t i ′′ of T of height i ′′ , a contradiction. So a S ( i ′′ +1 , j,t − S ( i ′′ , j,t for every i ′′ ∈ [0 , i ′ − . Hence a S ( i ′ , j,t . On theother hand, since t ∈ V ( T j,t ′ ) − ∂T j,t ′ and a ∈ V ( P ) ⊆ W ( i ′ ,ℓ +1 ,k )0 , we know a ∈ S ( i ′ +1 , j,t . Since X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t , we know a X t . Recall that b W ( i ′′ )4 for every i ′′ ∈ I ∗ .Since ab ∈ E ( G ) and V ( P ) ⊆ I j , we know for every i ′′ ∈ I ∗ , a ∈ X ∂T j,ti ′′ for some node t i ′′ of T of height i ′′ . So for every i ′′ ∈ I ∗ , there exists q i ′′ ∈ ∂T j,t i ′′ such that a ∈ X q i ′′ ∩ I j , where t i ′′ is a node of T of height i ′′ . If there exists i ′′ ∈ I ∗ with q i ′′ ∈ V ( T t ′ ) − { t ′ } , then t i ′′ is the nodeof T of height of i ′′ with t ′ ∈ V ( T t i ′′ ) , and since q ∈ ∂T j,t ′ and q i ′′ ∈ V ( T t ′ ) − { t ′ } , no node in ∂T j,t i ′′ is contained in the path in T from t i ′′ to the parent of q for every i ′′ ∈ I ∗ , so we have b ∈ W ( i ′′ )4 since b ∈ W ( i ′ )4 , a contradiction. Hence q i ′′ V ( T t ′ ) − { t ′ } for every i ′′ ∈ I ∗ . Since I ∗ = ∅ , q i ′′ V ( T t ′ ) − { t ′ } for some i ′′ ∈ I ∗ , so a ∈ X t ′ ∩ X t , since a ∈ X q i ′′ ∩ X V ( T t ) . Thiscontradicts that a X t .Hence z ∈ D ( i ′ ,ℓ, . Since z D ( i ′ ,ℓ, − X t ′ , z ∈ X t ′ . So there does not exist q ′ ∈ V ( T t ′ ) −{ t ′ } such that z ∈ X q ′ ∩ I j and q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′′ ,ℓ ′ )3 for some i ′′ ∈ [0 , i ′ − and ℓ ′ ∈ [0 , | V ( T ) | ] . Since ℓ = 0 , W ( i ′ ,ℓ − = W ( i ′ , − ⊆ X t ′ . Since t ′ is an ancestor of t and z ∈ X q , z ∈ X t ′ ∩ X t . Since z ∈ X q ∩ I j and q ∈ V ( T t ) − { t } and q is a witness for X q ∩ I j ⊆ W ( i ′ ,ℓ )3 ,and t ∈ V ( T j,t ′ ) − ∂T j,t ′ , we know z ∈ S ( i ′ +1 , j,t . Since X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t ,45 ∈ S ( i ′ , j,t . So there exist i z ∈ [0 , i ′ − , a node t z of T with σ T ( t z ) = i z with t ∈ V ( T j,t z ) − ∂T j,t z , ℓ z ∈ [0 , | V ( T ) | ] , and q z ∈ V ( T t ) such that z ∈ X q z ∩ I j and q z is a witness for X q z ∩ I j ⊆ W ( i z ,ℓ z )3 .But q z ∈ V ( T t ) ⊆ V ( T t ′ ) − { t ′ } . So q z is a node q ′ ∈ V ( T t ′ ) − { t ′ } such that z ∈ X q ′ ∩ I j and q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′′ ,ℓ ′ )3 for some i ′′ ∈ [0 , i ′ − and ℓ ′ ∈ [0 , | V ( T ) | ] , a contradiction.Therefore q V ( T t ) −{ t } . So ( W ( i ′ ,ℓ )3 − W ( i ′ ,ℓ − ) ∩ I j ∩ X V ( T t ) ⊆ ( W ( i ′ ,ℓ )3 − W ( i ′ ,ℓ − ) ∩ I j ∩ X t by the choice of q . Hence Y ( i ′ ,ℓ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 , ∩ X V ( T t ) ∩ I j − X t . Since Y ( i ′ ,ℓ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t , there exists k ′ ∈ [0 , s + 1] such that Y ( i ′ ,ℓ +1 ,k ′ +1) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,k ′ ) ∩ X V ( T t ) ∩ I j − X t . So W ( i ′ ,ℓ +1 ,k ′ )2 ∩ X V ( T t ) ∩ I j − X t = ∅ .Hence t ∈ V ( T j,t ′ ) − ∂T j,t ′ , and there exists x ∈ W ( i ′ ,ℓ +1 ,k ′ )0 ∩ X t ∩ I j = ∅ such that thereexists a monochromatic path Q x of color k ′ + 1 in G [ Y ( i ′ ,ℓ +1 ,k ′ ) ∩ W ( i ′ )4 ] from x to W ( i ′ ,ℓ )3 andinternally disjoint from X t ∪ W ( i ′ ,ℓ )3 , and there exists a monochromatic path Q ′ x of color k ′ + 1 with Q x ⊆ Q ′ x ⊆ W ( i ′ ,ℓ +1 ,k ′ )0 in G [ Y ( i ′ ,ℓ +1 ,k ′ ) ∩ W ( i ′ )4 ] from W ( i ′ ,ℓ )3 to a vertex x ′ and internallydisjoint from W ( i ′ ,ℓ )3 , and there exists x ′′ ∈ N G ( x ′ ) ∩ ( Y ( i ′ ,ℓ +1 ,k ′ +1) − Y ( i ′ ,ℓ +1 ,k ′ ) ) ∩ X V ( T t ) − X t = ∅ with x ′′ ∈ A L ( i ′ ,ℓ +1 ,k ′ ) ( x ′ ) and c ( x ′′ ) = c ( x ′ ) = c ( x ) = k ′ + 1 . Note that x ∈ W ( i ′ ,ℓ +1 ,k ′ )0 ⊆ D ( i ′ ,ℓ,k ′ +1) ⊆ D ( i ′ ,ℓ,s +2) . Since ℓ = 0 , X t ∩ I j ∩ D ( i ′ ,ℓ,s +2) = X t ∩ I j ∩ D ( i ′ ,ℓ, = X t ∩ I j ∩ D ( i ′ , , = X t ∩ I j ∩ D ( i ′ − , | V ( T ) | ,s +2) . So x ∈ D ( i ′ − , | V ( T ) | ,s +2) .Let I ′ = { i ′′ ∈ [0 , i ′ − 1] : x ∈ S | V ( T ) | ℓ ′ =0 W ( i ′′ ,ℓ ′ +1 ,k ′ )0 } . Since x ∈ D ( i ′ − , | V ( T ) | ,s +2) , I ′ = ∅ .So for every i ′′ ∈ I ′ , there exist t ′ i ′′ ∈ V ( T ) with σ T ( t ′ t ′′ ) = i ′′ , ℓ i ′′ ∈ [0 , | V ( T ) | ] , a witness q ′ i ′′ ∈ { t ′ i ′′ } ∪ ∂T j,t ′ i ′′ for X q ′ i ′′ ∩ I j ⊆ W ( i ′′ ,ℓ i ′′ )3 , and a monochromatic path P ′ i ′′ of color k ′ + 1 in G [ Y ( i ′′ ,ℓ i ′′ +1 ,k ′ ) ∩ W ( i ′′ )4 ] from X q ′ i ′′ ∩ I j to x . Note that if q ′ i ′′ is in the path in T from t ′ to t ,then q ′ i ′′ = t ′ , since t ∈ V ( T t ′ ,j ) − ∂T t ′ ,j .Suppose V ( Q ′ x ) ∪ { x ′′ } ⊆ W ( i ′′ )4 for some i ′′ ∈ I ′ . If V ( Q ′ x ) ⊆ Y ( i ′′ ,ℓ i ′′ +1 ,k ′ ) , then P ′ i ′′ ∪ Q ′ x is a connected monochromatic subgraph of color k ′ + 1 contained in G [ Y ( i ′′ ,ℓ i ′′ +1 ,k ′ ) ∩ W ( i ′′ )4 ] intersecting X q ′ i ′′ ∩ I j ⊆ W ( i ′′ ,ℓ i ′′ )3 and x ′ , so x ′′ ∈ W ( i ′′ ,ℓ i ′′ +1 ,k ′ )2 ⊆ Y ( i ′′ ,ℓ i ′′ +1 ,k ′ +1) ⊆ Y ( i ′ , , , contradicting that x ′′ ∈ Y ( i ′ ,ℓ +1 ,k ′ +1) − Y ( i ′ ,ℓ +1 ,k ′ ) . So V ( Q ′ x ) Y ( i ′′ ,ℓ i ′′ +1 ,k ′ ) . But thiscontradicts Claim 4.12.1, since P ′ i ′′ ∪ Q ′ x is a connected c -monochromatic subgraph containedin G [ W ( i ′′ )4 ] intersecting W ( i ′′ ,ℓ i ′′ )3 .Hence V ( Q ′ x ) ∪ { x ′′ } 6⊆ W ( i ′′ )4 for every i ′′ ∈ I ′ .Suppose that x ∈ S ( i ′ , j,t . So there exists i ′′ ∈ [0 , i ′ − such that x ∈ S ( i ′′ +1 , j,t − S ( i ′′ , j,t ⊆ S | V ( T ) | ℓ ′ =0 W ( i ′′ ,ℓ ′ +1 ,k ′ )0 (since c ( x ) = k ′ + 1 ) and t ∈ V ( T j,t ′′ ) − ∂T j,t ′′ , where t ′′ is the node of T with σ T ( t ′′ ) = i ′′ . Hence i ′′ ∈ I ′ and no node in ∂T j,t ′′ is in the path in T from t ′′ to theparent of t . Since i ′′ < i ′ < i and t ∈ V ( T j,t ′′ ) − ∂T j,t ′′ , for every q ∗ ∈ ∂T j,t ′ , we know that ∂T j,t ′′ is disjoint from the path in T from t ′ to the parent of q ∗ . Hence for every q ∗ ∈ ∂T j,t ′ , ∂T j,t ′′ is disjoint from the path in T from t ′′ to the parent of q ∗ . So W ( i ′ )4 ⊆ W ( i ′′ )4 . Hence V ( Q ′ x ) ∪ { x ′′ } ⊆ W ( i ′ )4 ⊆ W ( i ′′ )4 , a contradiction.Hence x S ( i ′ , j,t . However, x ∈ W ( i ′ ,ℓ +1 ,k ′ )0 and t ∈ V ( T j,t ′ ) − ∂T j,t ′ , so x ∈ S ( i ′ +1 , j,t .Therefore, X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t , a contradiction. This proves the claim. Claim 4.12.3. Let i, i ′ ∈ N with i ′ < i . Let t, t ′ be nodes of T with σ T ( t ) = i and σ T ( t ′ ) = i ′ .Let j ∈ [ | V | − . Let ℓ ∈ [ − , | V ( T ) | ] such that there exists a witness q ∈ ∂T j,t ′ ∩ V ( T t ) − { t } for X q ∩ I j ⊆ W ( i ′ ,ℓ +1)3 and X q ∩ I j W ( i ′ ,ℓ )3 . Then either: | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | , or • | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | , or • for every c -monochromatic path P in G [ Y ( i ′ ,ℓ +1 ,s +2) ∩ I j ∩ W ( i ′ )4 ] from W ( i ′ ,ℓ )3 to X q ∩ I j − (( D ( i ′ ,ℓ +1 , − X t ′ ) ∪ { v ∈ D ( i ′ ,ℓ +1 , ∩ X t ′ ∩ X q ′ ∩ I j : q ′ ∈ V ( T t ′ ) − { t ′ } ,q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′′ ,ℓ ′ )3 for some i ′′ ∈ [0 , i ′ − and ℓ ′ ∈ [0 , | V ( T ) | ] } ) and internally disjoint from X t ′ ∪ X ∂T j,t ′ , we have V ( P ) ⊆ X V ( T t ) − X t .Proof. Note that t ′ is an ancestor of t , since q ∈ ∂T j,t ′ ∩ V ( T t ) − { t } . We may assume that | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | and | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | , for otherwisewe are done. So X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t and X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t .Since there exists a witness q ∈ ∂T j,t ′ ∩ V ( T t ) − { t } for X q ∩ I j ⊆ W ( i ′ ,ℓ +1)3 and X q ∩ I j W ( i ′ ,ℓ )3 , there exists a c -monochromatic path P in G [ Y ( i ′ ,ℓ +1 ,s +2) ∩ I j ∩ W ( i ′ )4 ] from W ( i ′ ,ℓ )3 to avertex z ∈ X q ∩ I j − (( D ( i ′ ,ℓ +1 , − X t ′ ) ∪ { v ∈ D ( i ′ ,ℓ +1 , ∩ X t ′ ∩ X q ′ ∩ I j : q ′ ∈ V ( T t ′ ) − { t ′ } ,q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′′ ,ℓ ′ )3 for some i ′′ ∈ [0 , i ′ − and ℓ ′ ∈ [0 , | V ( T ) | ] } ) and internally disjoint from X t ′ ∪ X ∂T j,t ′ . Furthermore, choose P such that V ( P ) X V ( T t ) − X t if possible. We may assume that V ( P ) X V ( T t ) − X t , for otherwise we are done.Let k ∈ [0 , s + 1] be the number such that P is of color k + 1 . By Claim 4.12.1, V ( P ) ⊆ Y ( i ′ ,ℓ +1 ,k ) . Since V ( P ) X V ( T t ) − X t and q ∈ V ( T t ) − { t } , there exists a vertex z ′ in V ( P ) ∩ X t such that the subpath of P between z and z ′ is contained in G [ X V ( T t ) ] and internally disjointfrom X t . If ℓ > , then let ℓ = ℓ ; if ℓ = − , then let ℓ = 0 . Since ℓ > ℓ and V ( P ) ⊆ Y ( i ′ ,ℓ +1 ,k ) , V ( P ) ⊆ Y ( i ′ ,ℓ +1 ,k ) . Since ℓ > , V ( P ) ⊆ W ( i ′ ,ℓ +1 ,k )0 . So z ′ ∈ S ( i ′ +1 , j,t ∩ X t ∩ I j .Since X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t , z ′ ∈ S ( i ′ , j,t ∩ X t ∩ I j . Hence there exists i ∈ [0 , i ′ − such that z ′ ∈ S ( i +1 , j,t − S ( i , j,t . Note that such i exists since S (0 , j,t = ∅ . Let t be the nodeof T with σ T ( t ) = i . Since z ′ ∈ S ( i +1 , j,t − S ( i , j,t , t ∈ V ( T j,t ) − ∂T j,t , and there exists ℓ z ′ ∈ [0 , | V ( T ) | ] such that z ′ ∈ W ( i ,ℓ z ′ +1 ,k )0 . Hence there exists a c -monochromatic path P z ′ in G [ Y ( i ,ℓ z ′ +1 ,k ) ∩ W ( i )4 ] from z ′ to W ( i ,ℓ z ′ )3 .Since t ∈ V ( T j,t ) − ∂T j,t , ∂T j,t is disjoint from the path in T from t to t ′ . So V ( T j,t ′ ) ⊆ V ( T j,t ) as i < i ′ . So P ∪ P z ′ is a c -monochromatic subgraph of color k + 1 containedin W ( i )4 containing z and intersecting W ( i ,ℓ z ′ )3 . By Claim 4.12.1, the path P z contained in P ∪ P z ′ from W ( i ,ℓ z ′ )3 to z satisfies V ( P z ) ⊆ Y ( i ,ℓ z ′ +1 ,k ) . Hence V ( P z ) ⊆ W ( i ,ℓ z ′ +1 ,k )0 . So z ∈ D ( i ,ℓ z ′ ,k +1) ⊆ D ( i ′ ,ℓ +1 , . Since z ∈ X q ∩ I j − ( D ( i ′ ,ℓ +1 , − X t ′ ) , z ∈ X t ′ . Since t belongsto the path in T between t ′ and q , z ∈ X t . Since q ∈ ∂T j,t ′ ∩ V ( T t ) − { t } , t ∈ V ( T j,t ′ ) − ∂T j,t .Since z ∈ X q ∩ I j , z ∈ S ( i ′ +1 , j,t . Since S ( i ′ +1 , j,t ∩ X t ∩ I j = S ( i ′ , j,t ∩ X t ∩ I j , z ∈ S ( i ′ , j,t . So thereexists i ′ ∈ [0 , i ′ − such that z ∈ S ( i ′ +1 , j,t − S ( i ′ , j,t . Note that i ′ exists since S (0 , j,t = ∅ .Let t ′ be the node of T with σ T ( t ′ ) = i ′ . Since z ∈ S ( i ′ +1 , j,t − S ( i ′ , j,t , t ∈ V ( T j,t ′ ) − ∂T j,t ′ and there exists q z ∈ V ( T t ) −{ t } such that z ∈ X q z ∩ I j and q z is a witness for X q z ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 for some ℓ ′ ∈ [0 , | V ( T ) | ] . Note that q z ∈ V ( T t ′ ) − { t ′ } since q z ∈ V ( T t ) − { t } . However, the47xistence of q z contradicts that z ∈ X q ∩ I j −{ v ∈ D ( i ′ ,ℓ +1 , ∩ X t ′ ∩ X q ′ ∩ I j : q ′ ∈ V ( T t ′ ) −{ t ′ } , q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′′ ,ℓ ′ )3 for some i ′′ ∈ [0 , i ′ − and ℓ ′ ∈ [0 , | V ( T ) | ] } . This provesthe claim. Claim 4.12.4. Let i, i ′ ∈ N with i ′ < i . Let t, t ′ be nodes of T with σ T ( t ) = i and σ T ( t ′ ) = i ′ .Let j ∈ [ | V | − . Let ℓ ∈ [ − , | V ( T ) | ] such that there exists no witness q ∈ ∂T j,t ′ ∩ V ( T t ) − { t } for X q ∩ I j ⊆ W ( i ′ ,ℓ +1)3 and X q ∩ I j W ( i ′ ,ℓ )3 . If Y ( i ′ ,ℓ +2 ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t , then either: • | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | , or • | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | .Proof. Since Y ( i ′ ,ℓ +2 ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t , t ′ is an ancestor of t . Suppose to the contrary that | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | and | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | . So X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t and X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t .Since there exists no witness q ∈ ∂T j,t ′ ∩ V ( T t ) − { t } for X q ∩ I j ⊆ W ( i ′ ,ℓ +1)3 and X q ∩ I j W ( i ′ ,ℓ )3 , W ( i ′ ,ℓ +1)3 ∩ X V ( T t ) ∩ I j − X t = W ( i ′ ,ℓ )3 ∩ X V ( T t ) ∩ I j − X t . This implies that Y ( i ′ ,ℓ +2 , ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t . Since Y ( i ′ ,ℓ +2 ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t , Y ( i ′ ,ℓ +2 , ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +2 ,s +2) ∩ X V ( T t ) ∩ I j − X t . Hencethere exists k ∈ [0 , s + 1] such that Y ( i ′ ,ℓ +2 ,k +1) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +2 ,k ) ∩ X V ( T t ) ∩ I j − X t .So there exist x ′ ∈ W ( i ′ ,ℓ +2 ,k )2 ∩ W ( i ′ )4 ∩ X V ( T t ) ∩ I j − ( X t ∪ Y ( i ′ ,ℓ +2 ,k ) ) , x ∈ N G ( x ′ ) ∩ W ( i ′ ,ℓ +2 ,k )0 anda monochromatic path Q in G [ Y ( i ′ ,ℓ +2 ,k ) ∩ W ( i ′ )4 ] of color k + 1 from x to W ( i ′ ,ℓ +1)3 , such that x ′ ∈ A L ( i ′ ,ℓ +2 ,k ) ( x ) . Note that x ∈ X V ( T t ) . Let q Q ∈ ∂T j,t ′ be a witness for X q Q ∩ I j ⊆ W ( i ′ ,ℓ +1)3 such that one end of Q is in X q Q ∩ I j . Since x ′ ∈ X V ( T t ) − X t , t ∂T j,t ′ .Suppose that q Q ∈ V ( T t ) . Since q Q ∈ ∂T j,t ′ , q Q ∈ V ( T t ) −{ t } . Since there exists no witness q ∈ ∂T j,t ′ ∩ V ( T t ) −{ t } for X q ∩ I j ⊆ W ( i ′ ,ℓ +1)3 and X q ∩ I j W ( i ′ ,ℓ )3 , we know X q Q ∩ I j ⊆ W ( i ′ ,ℓ )3 .So by Claim 4.12.1, V ( Q ) ⊆ W ( i ′ ,ℓ +1 ,k )0 and x ′ ∈ W ( i ′ ,ℓ +1 ,k )2 ⊆ Y ( i ′ ,ℓ +1 ,k ) ⊆ Y ( i ′ ,ℓ +2 ,k ) , acontradiction.So q Q V ( T t ) . Hence there exists a vertex x ∗ ∈ X t ∩ I j such that the subpath of Q between x ∗ and x is contained in G [ X V ( T t ) ] . Since x ∗ ∈ X t ∩ V ( Q ) ⊆ X t ∩ I j ∩ W ( i ′ ,ℓ +2 ,k )0 and t ∈ V ( T j,t ′ ) − ∂T j,t ′ , x ∗ ∈ S ( i ′ +1 , j,t ∩ X t ∩ I j . Since X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t , there exists i ∈ [0 , i ′ − such that x ∗ ∈ S ( i +1 , j,t − S ( i , j,t . So the node t of T with σ T ( t ) = i satisfiesthat t ∈ V ( T j,t ) − ∂T j,t , and there exists ℓ x ∗ ∈ [0 , | V ( T ) | ] such that x ∗ ∈ W ( i ,ℓ x ∗ +1 ,k )0 , since c ( x ∗ ) = k + 1 . Hence there exists a monochromatic path Q x ∗ in G [ Y ( i ,ℓ x ∗ +1 ,k ) ∩ W ( i )4 ] from x ∗ to W ( i ,ℓ x ∗ )3 .Since t ∈ V ( T j,t ) − ∂T j,t , ∂T j,t is disjoint from the path in T from t to t ′ . Since i < i ′ , V ( T j,t ′ ) ⊆ V ( T j,t ) . So Q ∪ Q x ∗ is a c -monochromatic subgraph of color k + 1 contained in W ( i )4 containing x and intersects W ( i ,ℓ x ∗ )3 . By Claim 4.12.1, the path Q x contained in Q ∪ Q x ∗ from W ( i ,ℓ x ∗ )3 to x satisfies V ( Q x ) ⊆ Y ( i ,ℓ x ∗ +1 ,k ) and A L ( i ,ℓx ∗ +1 ,k +1) ( V ( Q x )) ∩ W ( i )4 = ∅ . But x ′ ∈ A L ( i ′ ,ℓ +2 ,k ) ( V ( Q x )) ∩ W ( i ′ )4 ⊆ A L ( i ,ℓx ∗ +1 ,k +1) ( V ( Q x )) ∩ W ( i )4 = ∅ , a contradiction. Thisproves the claim. 48 laim 4.12.5. Let i, i ′ ∈ N with i ′ < i , and let t be the node of T with σ T ( t ) = i . Let j ∈ [ | V | − . If Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t , then either: • Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t , or • | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | , or • | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | .Proof. We may assume that Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t ,for otherwise we are done. Since Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t , Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t . So there exists a minimum ℓ ∈ [0 , | V ( T ) | − such that Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +2 ,s +2) ∩ X V ( T t ) ∩ I j − X t .By the minimality of ℓ , Y ( i ′ ,ℓ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ ,ℓ +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t .Let t ′ be the node of T with σ T ( t ′ ) = i ′ . Since Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t , t ∈ V ( T j,t ′ ) − ∂T j,t ′ .Suppose to the contrary that | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | and | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | . So X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t and X t ∩ I j ∩ S ( i ′ +1 , j,t = X t ∩ I j ∩ S ( i ′ , j,t .We first suppose that there exist ℓ ∈ [ − , ℓ ] and a witness q ∈ ∂T j,t ′ ∩ V ( T t ) for X q ∩ I j ⊆ W ( i ′ ,ℓ +1)3 and X q ∩ I j W ( i ′ ,ℓ )3 . Choose ℓ so that ℓ is as small as possible. Sothere exists a c -monochromatic path P in G [ Y ( i ′ ,ℓ +1 ,s +2) ∩ I j ∩ W ( i ′ )4 ] from W ( i ′ ,ℓ )3 to a node z ∈ X q ∩ I j − (( D ( i ′ ,ℓ +1 , − X t ′ ) ∪ { v ∈ D ( i ′ ,ℓ +1 , ∩ X t ′ ∩ X q ′ ∩ I j : q ′ ∈ V ( T t ′ ) − { t ′ } , q ′ is awitness for X q ′ ∩ I j ⊆ W ( i ′′ ,ℓ ′ )3 for some i ′′ ∈ [0 , i ′ − and ℓ ′ ∈ [0 , | V ( T ) | ] } ) internally disjointfrom X t ′ ∪ X ∂T j,t ′ . Since t ∂T j,t ′ , by Claim 4.12.3, V ( P ) ⊆ X V ( T t ) − X t . In particular, ℓ > .Let k ∈ [0 , s + 1] be the number such that P is of color k + 1 . If V ( P ) intersects W ( i ′ ,ℓ − ,then by Claim 4.12.1, V ( P ) ⊆ Y ( i ′ ,ℓ ,k ) , so q is a witness for X q ∩ I j ⊆ W ( i ′ ,ℓ )3 , a contradiction.So V ( P ) ∩ W ( i ′ ,ℓ − = ∅ . Then there exists a witness q ′ ∈ V ( T t ) − { t } for X q ′ ∩ I j ⊆ W ( i ′ ,ℓ )3 with X q ′ ∩ I j W ( i ′ ,ℓ − such that V ( P ) is from X q ′ to X q , contradicting the minimality of ℓ . Therefore, there do not exist ℓ ∈ [ − , ℓ ] and a witness q ∈ ∂T j,t ′ ∩ V ( T t ) for X q ∩ I j W ( i ′ ,ℓ +1)3 and X q ∩ I j W ( i ′ ,ℓ )3 . In particular, there exists no witness q ∈ ∂T j,t ′ ∩ V ( T t ) for X q ∩ I j ⊆ W ( i ′ ,ℓ +1)3 and X q ∩ I j W ( i ′ ,ℓ )3 . By Claim 4.12.4, either | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | , or | X t ∩ I j ∩ S ( i ′ +1 , j,t | > | X t ∩ I j ∩ S ( i ′ , j,t | , a contradiction. This proves theclaim. Claim 4.12.6. Let i ∈ N , and let t be the node of T with σ T ( t ) = i . Let j ∈ [ | V | − . Let S := { i ′ ∈ [0 , i − 1] : Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t } . Then | S | w .Proof. Let S := { i ′ ∈ [0 , i − 1] : Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t = Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t } ,and let S := { i ′ ∈ [0 , i − 1] : | X t ∩ I j ∩ S ( i ′ +1 , j,t | + | X t ∩ I j ∩ S ( i ′ +1 , j,t | + | X t ∩ I j ∩ D ( i ′ , ,s +2) | > | X t ∩ I j ∩ S ( i ′ , j,t | + | X t ∩ I j ∩ S ( i ′ , j,t | + | X t ∩ I j ∩ D ( i ′ , , |} . By Claim 4.12.5, S ⊆ S ∪ S . ByClaim 4.12.2, S ⊆ S . So S ⊆ S . Since | X t ∩ I j | w , | S | | S | w . Claim 4.12.7. Let t ∈ V ( T ) and j ∈ [ | V | − . Let ( Y ′ , L ′ ) be a V -standard pair and let Z ⊆ Y ′ . Then A L ′ ( Z ) ⊆ N > sG [ Z ] , A L ′ ( Z ) ∩ X V ( T j,t ) ⊆ N > sG ( Z ∩ X V ( T j,t ) ) ∪ X ∂T j,t ∪ X t , and | A L ′ ( Z ) ∩ X V ( T j,t ) | f ( | Z ∩ X V ( T j,t ) | ) + | X ∂T j,t | + | X t | . roof. It is obvious that A L ′ ( Z ) ⊆ N > sG [ Z ] since ( Y ′ , L ′ ) is a V -standard pair. Since ( Y ′ , L ′ ) is a V -standard pair, A L ′ ( Z ) ∩ X V ( T j,t ) ⊆ N > sG ( Z ) ∩ X V ( T j,t ) ⊆ N > sG ( Z ∩ X V ( T j,t ) ) ∪ ( X ∂T j,t ∪ X t ) .So | A L ′ ( Z ) ∩ X V ( T j,t ) | | N > sG ( Z ∩ X V ( T j,t ) ) | + | X ∂T j,t | + | X t | f ( | Z ∩ X V ( T j,t ) | ) + | X ∂T j,t | + | X t | by Lemma 3.6. Claim 4.12.8. Let i, i ′ ∈ N with i ′ < i , and let t and t ′ be nodes of T with σ T ( t ) = i and σ T ( t ′ ) = i ′ . Let j ∈ [ | V | − . Then the following hold: • | Y ( i ′ , − , ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | w . • | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | w .Proof. We may assume that t ∈ V ( T t ′ ) , for otherwise, X V ( T t ) ∩ X V ( T j,t ′ ) = ∅ and we are done.Let F j,p := F j, if t ′ = r ∗ ; let p be the parent of t ′ if t ′ = r ∗ . By definition, F j,p = F ′ j,p ∪ S T ′ F j,T ′ , where the union is over all F ′ j,p ∩ V ( T t ′ ) -parts T ′ of T t ′ containing t ′ , and each F j,T ′ is a ( T ′ , X X V ( T ′ ) ∩ I j , ( Y ( i ′ , − , ∪ X ∂T ′ ) ∩ X V ( T ′ ) ∩ I j ) -fence. So for each F j,p ∩ V ( T t ′ ) -part T ′ of T t ′ , | ( Y ( i ′ , − , ∩ I j ∩ X V ( T ′ ) ) ∪ ( X ∂T ′ ∩ I j ) | w . By definition, T j,t ′ = S T ′ T ′ , where the unionis over all F j,p ∩ V ( T t ′ ) -parts T ′ of T t ′ containing t ′ . Since t ∈ V ( T t ′ ) −{ t ′ } , there exists at mostone F j,p ∩ V ( T t ′ ) -part of T t ′ containing both t ′ and t . If there exists an F j,p ∩ V ( T t ′ ) -part of T t ′ containing both t ′ and t , then denote it by T ∗ ; otherwise let T ∗ := ∅ . So X V ( T t ) ∩ X V ( T j,t ′ ) ⊆ X V ( T ∗ ) . Hence | Y ( i ′ , − , ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | | Y ( i ′ , − , ∩ I j ∩ X V ( T ∗ ) | w . Similarly, | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | | X ∂T j,t ′ ∩ X V ( T j,t ′ ) ∩ X V ( T t ) ∩ I j | | X ∂T j,t ′ ∩ X V ( T ∗ ) ∩ I j | | X ∂T ∗ ∩ I j | w . Claim 4.12.9. Let i, i ′ ∈ N with i ′ < i , and let t and t ′ be the nodes of T with σ T ( t ) = i and σ T ( t ′ ) = i ′ . Let j ∈ [ | V | − . Then | Y ( i ′ , , ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g ( w ) .Proof. For every k ∈ [ w ] , let q k ∈ [0 , s + 1] be the number such that the color of M ( t ′ ) j,k is q k + 1 . So for every k ∈ [ w ] , ( Y ( i ′ , − ,k +1) − Y ( i ′ , − ,k ) ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) − X t ⊆ A L ( i ′ , − ,k,qk ) ( W ( i ′ , − ,k,q k )1 ) ∩ Z t ′ ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) − X t ⊆ ( N > sG ( W ( i ′ , − ,k,q k )1 ∩ X V ( T j,t ′ ) ) ∪ X ∂T j,t ′ ∪ X t ′ ) ∩ Z t ′ ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) − X t , where the last inclusion follows from Claim 4.12.7. Note that for every k ∈ [ w ] , sinceevery vertex in W ( i ′ , − ,k,q k )1 ∩ I j is connected by a monochromatic path to I ◦ j , we know N G [ W ( i ′ , − ,k,q k )1 ∩ I j ] ⊆ I j . So for every k ∈ [ w ] , ( N > sG ( W ( i ′ , − ,k,q k )1 ∩ X V ( T j,t ′ ) ) ∪ X ∂T j,t ′ ∪ X t ′ ) ∩ Z t ′ ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) − X t ⊆ ( N > sG ( W ( i ′ , − ,k,q k )1 ∩ X V ( T j,t ′ ) ) ∩ Z t ′ ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) − X t ) ∪ ( X ∂T j,t ′ ∩ I j ∩ X V ( T t ) ) ⊆ ( N > sG ( W ( i ′ , − ,k,q k )1 ∩ X V ( T j,t ′ ) ∩ I j ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) − X t ) ∪ ( X ∂T j,t ′ ∩ I j ∩ X V ( T t ) ) ⊆ N > sG ( W ( i ′ , − ,k,q k )1 ∩ X V ( T j,t ′ ) ∩ I j ∩ X V ( T t ) ) ∪ ( X ∂T j,t ′ ∩ I j ∩ X V ( T t ) ) ⊆ N > sG ( Y ( i ′ , − ,k ) ∩ X V ( T j,t ′ ) ∩ I j ∩ X V ( T t ) ) ∪ ( X ∂T j,t ′ ∩ I j ∩ X V ( T t ) ) . k ∈ [ w ] , | Y ( i ′ , − ,k +1) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | | Y ( i ′ , − ,k ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | + | X t ∩ I j | + | ( Y ( i ′ , − ,k +1) − Y ( i ′ , − ,k ) ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) − X t | | Y ( i ′ , − ,k ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | + | X t ∩ I j | + f ( | Y ( i ′ , − ,k ) ∩ X V ( T j,t ′ ) ∩ I j ∩ X V ( T t ) | ) + | X ∂T j,t ′ ∩ I j ∩ X V ( T t ) | = f ( | Y ( i ′ , − ,k ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + | X t ∩ I j | + | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | f ( | Y ( i ′ , − ,k ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + 2 w , where the last inequality follows from Claim 4.12.8.By Claim 4.12.8, | Y ( i ′ , − , ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | w = g (0) . So it is easy to verify thatfor every k ∈ [ w ] , | Y ( i ′ , − ,k ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g ( k ) by induction on k . Therefore, | Y ( i ′ , , ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | = | Y ( i ′ , − ,w ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g ( w ) . Claim 4.12.10. Let i, i ′ ∈ N with i ′ i , and let t and t ′ be the nodes of T with σ T ( t ) = i and σ T ( t ′ ) = i ′ . Let j ∈ [ | V | − . Let a, b be integers such that I j = S bα = a V α . Then thefollowing hold: • For every k ∈ [0 , s + 1] , | Y ( i ′ , ,k +1) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | f ( | Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | + 2 w . • If i ′ < i , then | Y ( i ′ , ,s +2) ∩ ( S b +1 α = a − V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g ( s + 2) . • If i ′ < i , then for every ℓ ∈ [0 , | V ( T ) | + 1] , | Y ( i ′ ,ℓ +1 ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g ( s + 2 , | Y ( i ′ ,ℓ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) . • If i ′ < i , then | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ ( S b +1 α = a − V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g (4 w ) .Proof. We may assume that t ∈ V ( T t ′ ) , for otherwise X V ( T t ) ∩ X V ( T j,t ′ ) = ∅ and we are done.We first prove Statement 1. For every k ∈ [0 , s + 1] , ( Y ( i ′ , ,k +1) − Y ( i ′ , ,k ) ) ∩ I j ∩ X V ( T t ) ⊆ W ( i ′ , ,k )2 ∩ I j ∩ X V ( T t ) ⊆ A L ( i ′ , ,k ) ( Y ( i ′ , ,k ) ) ∩ Z t ′ ∩ I j ∩ X V ( T t ) . ( Y ( i ′ , ,k +1) − Y ( i ′ , ,k ) ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ) ⊆ A L ( i ′ , ,k ) ( Y ( i ′ , ,k ) ) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,t ) . ByClaim 4.12.7, A L ( i ′ , ,k ) ( Y ( i ′ , ,k ) ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) − X t ⊆ A L ( i ′ , ,k ) ( Y ( i ′ , ,k ) ∩ X V ( T t ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α )) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+ k +1 V α ) ∩ X V ( T t ) − X t , and A L ( i ′ , ,k ) ( Y ( i ′ , ,k ) ∩ X V ( T t ) ) ∩ X V ( T j,t ′ ) ⊆ N > sG ( Y ( i ′ , ,k ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) ) ∪ X ∂T j,t ′ ∪ X t ′ . Therefore, ( Y ( i ′ , ,k +1) − Y ( i ′ , ,k ) ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) ⊆ ( A L ( i ′ , ,k ) ( Y ( i ′ , ,k ) ) ∩ X V ( T j,t ′ ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) − X t ) ∪ ( X t ∩ I j ) ⊆ (cid:16) A L ( i ′ , ,k ) ( Y ( i ′ , ,k ) ∩ X V ( T t ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α )) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) − X t (cid:17) ∪ ( X t ∩ I j ) ⊆ (cid:16) ( N > sG ( Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) ) ∪ X ∂T j,t ′ ∪ X t ′ ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+ k +1 V α ) ∩ X V ( T t ) − X t (cid:17) ∪ ( X t ∩ I j ) , where the last inequality follows from Claim 4.12.7. Hence | ( Y ( i ′ , ,k +1) − Y ( i ′ , ,k ) ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | | N > sG ( Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+ k +1 V α ) ∩ X V ( T t ) − X t | + | ( X ∂T j,t ′ ∪ X t ′ ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+ k +1 V α ) ∩ X V ( T t ) − X t | + | X t ∩ I j | f ( | Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | + | X t ′ ∩ I j | + | X t ∩ I j | f ( | Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | + 2 w . So | Y ( i ′ , ,k +1) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | = | Y ( i ′ , ,k ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | + | ( Y ( i ′ , ,k +1) − Y ( i ′ , ,k ) ) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | | Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | + f ( | Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | + 2 w = f ( | Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | + 2 w . This proves Statement 1.Now we prove Statement 2 of this claim. Assume i ′ < i . By Claim 4.12.8 and Statement 1of this claim, for every k ∈ [0 , s + 1] , | Y ( i ′ , ,k +1) ∩ ( b +( s +3) − ( k +1) [ α = a − ( s +3)+( k +1) V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | f ( | Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + | X ∂T j,t ′ ∩ X V ( T t ) ∩ I j | + 2 w f ( | Y ( i ′ , ,k ) ∩ ( b +( s +3) − k [ α = a − ( s +3)+ k V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | ) + 3 w . Then it is easy to verify that for every k ∈ [0 , s + 2] , | Y ( i ′ , ,k ) ∩ ( S b +( s +3) − kα = a − ( s +3)+ k ) V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g ( k ) by induction on k . (Note that the base case k = 0 follows from Claim 4.12.9.)Then Statement 2 follows from the case k = s + 2 .Now we prove Statement 3 of this claim. Note that for every ℓ ∈ [0 , | V ( T ) | ] , | Y ( i ′ ,ℓ +1 , ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | | Y ( i ′ ,ℓ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | + | W ( i ′ ,ℓ )3 ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | | Y ( i ′ ,ℓ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | + | X ∂T j,t ′ ∩ I j ∩ X V ( T t ) | | Y ( i ′ ,ℓ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | + w = g (0 , | Y ( i ′ ,ℓ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | ) by Claim 4.12.8. For every ℓ ∈ [0 , | V ( T ) | ] and k ∈ [0 , s + 1] , by Claim 4.12.7, we know W ( i ′ ,ℓ +1 ,k )2 ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) = A L ( i ′ ,ℓ +1 ,k ) ( W ( i ′ ,ℓ +1 ,k )0 ) ∩ W ( i ′ )4 ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) ⊆ A L ( i ′ ,ℓ +1 ,k ) ( W ( i ′ ,ℓ +1 ,k )0 ) ∩ X V ( T j,t ′ ) ∩ I j ⊆ ( A L ( i ′ ,ℓ +1 ,k ) ( W ( i ′ ,ℓ +1 ,k )0 ∩ X V ( T j,t ′ ) ) ∪ X ∂T j,t ′ ∪ X t ′ ) ∩ X V ( T j,t ′ ) ∩ I j ⊆ ( A L ( i ′ ,ℓ +1 ,k ) ( W ( i ′ ,ℓ +1 ,k )0 ∩ I j ∩ X V ( T j,t ′ ) ) ∪ X ∂T j,t ′ ∪ X t ′ ) ∩ X V ( T j,t ′ ) ∩ I j ⊆ ( N > sG ( W ( i ′ ,ℓ +1 ,k )0 ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ) ∪ X ∂T j,t ′ ∪ X t ′ ) ∩ X V ( T j,t ′ ) ∩ I j . So for every ℓ ∈ [0 , | V ( T ) | ] and k ∈ [0 , s + 1] , ( Y ( i ′ ,ℓ +1 ,k +1) − Y ( i ′ ,ℓ +1 ,k ) ) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) ⊆ W ( i ′ ,ℓ +1 ,k )2 ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) ⊆ ( N > sG ( W ( i ′ ,ℓ +1 ,k )0 ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ) ∪ X ∂T j,t ′ ∪ X t ′ ) ∩ X V ( T j,t ′ ) ∩ I j ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) ⊆ ( N > sG ( W ( i ′ ,ℓ +1 ,k )0 ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) ) ∪ X t ∪ X ∂T j,t ′ ∪ X t ′ ) ∩ X V ( T j,t ′ ) ∩ I j ∩ X V ( T t ) ⊆ N > sG [ Y ( i ′ ,ℓ +1 ,k ) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) ] ∪ ( X t ∩ I j ) ∪ ( X ∂T j,t ′ ∩ I j ∩ X V ( T t ) ) ∪ ( X t ′ ∩ I j ) . Hence for every ℓ ∈ [0 , | V ( T ) | ] and k ∈ [0 , s + 1] , | ( Y ( i ′ ,ℓ +1 ,k +1) − Y ( i ′ ,ℓ +1 ,k ) ) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) | | N > sG [ Y ( i ′ ,ℓ +1 ,k ) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) ] | + | X t ∩ I j | + | X ∂T j,t ′ ∩ I j ∩ X V ( T t ) | + | X t ′ ∩ I j | f ( Y ( i ′ ,ℓ +1 ,k ) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) ) + 3 w by Claim 4.12.8. So for every ℓ ∈ [0 , | V ( T ) | ] and k ∈ [0 , s + 1] , | Y ( i ′ ,ℓ +1 ,k +1) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | f ( Y ( i ′ ,ℓ +1 ,k ) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) ) + 3 w . Then for every ℓ ∈ [0 , | V ( T ) | ] , it is easy to verify by induction on k ∈ [0 , s + 2] that | Y ( i ′ ,ℓ +1 ,k ) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | g ( k, | Y ( i ′ ,ℓ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | ) . Hence | Y ( i ′ ,ℓ +1 ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | g ( s + 2 , | Y ( i ′ ,ℓ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | ) . Now we prove Statement 4 of this claim. Let S := { ℓ ∈ [0 , | V ( T ) | ] : | Y ( i ′ ,ℓ +1 ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) | > | Y ( i ′ ,ℓ,s +2) ∩ ( b +1 [ α = a − V α ) ∩ X V ( T t ) |} . For every ℓ ∈ S , either ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ X t ∩ ( S b +1 α = a − V α ) = ∅ , or ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ ( S b +1 α = a − V α ) ∩ X V ( T t ) − X t = ∅ . Note that ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ S b +1 α = a − V α ⊆ ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ W ( i ′ )4 ∩ S b +1 α = a − V α ⊆ ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ I j . So for every ℓ ∈ S , either ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ X t ∩ ( S b +1 α = a − V α ) = ∅ , or ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ I j ∩ X V ( T t ) − X t = ∅ .Let S := { ℓ ∈ [0 , | V ( T ) | ] : ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ X t ∩ ( b +1 [ α = a − V α ) = ∅} S := { ℓ ∈ [0 , | V ( T ) | ] : ( Y ( i ′ ,ℓ +1 ,s +2) − Y ( i ′ ,ℓ,s +2) ) ∩ I j ∩ X V ( T t ) − X t = ∅} . So S ⊆ S ∪ S . Note that | S | | X t ∩ ( S b +1 α = a − V α ) | w . Let S := { ℓ ∈ [0 , | V ( T ) | ] : ( W ( i ′ ,ℓ )3 − W ( i ′ ,ℓ − ) ∩ I j ∩ X V ( T t ) = ∅} . Since W ( i ′ ,ℓ )3 ∩ I j ∩ X V ( T t ) ⊆ X ∂T j,t ′ ∩ I j ∩ X V ( T t ) for every ℓ ∈ [0 , | V ( T ) | ] , we know | S | | X ∂T j,t ′ ∩ I j ∩ X V ( T t ) | w by Claim 4.12.8 since i ′ < i . Let S := { ℓ ∈ [0 , | V ( T ) | ] : | X t ∩ I j ∩ S ( i ′ +1 , j,t | + | X t ∩ I j ∩ S ( i ′ +1 , j,t | | X t ∩ I j ∩ S ( i ′ , j,t | + | X t ∩ I j ∩ S ( i ′ , j,t |} . For every ℓ ∈ S − S , we know ( W ( i ′ ,ℓ )3 − W ( i ′ ,ℓ − ) ∩ I j ∩ X V ( T t ) = ∅ , so there exists nowitness q ∈ ∂T j,t ′ ∩ V ( T t ) − { t } for X q ∩ I j ⊆ W ( i ′ ,ℓ )3 and X q ∩ I j W ( i ′ ,ℓ − , and hence ℓ ∈ S by Claim 4.12.4. So S ⊆ S ∪ S ⊆ S ∪ S ∪ S . Therefore, | S | | S | + | S | + | S | w + w + 2 w = 4 w .By Statements 2 and 3 of this claim, it is easy to verify by induction on k ∈ [ | S | ] that | Y ( i ′ ,ℓ k ,s +2) ∩ ( S b +1 α = a − V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g ( k ) , where we denote the elements of S by ℓ < ℓ < · · · < ℓ | S | . Therefore, | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ ( S b +1 α = a − V α ) ∩ X V ( T t ) ∩ X V ( T j,t ′ ) | g ( | S | ) g (4 w ) . Claim 4.12.11. Let i ∈ N and let t be a node of T with σ T ( t ) = i . Let j ∈ [ |V| − . If i ′ isan integer in [0 , i − such that either | Y ( i ′ , , ∩ I ◦ j ∩ X V ( T t ) − X t | 6 = | Y ( i ′ , ,s +2) ∩ I ◦ j ∩ X V ( T t ) − X t | or | Y ( i ′ , − , ∩ I ◦ j ∩ X V ( T t ) − X t | 6 = | Y ( i ′ , − ,w ) ∩ I ◦ j ∩ X V ( T t ) − X t | , then | S ( i ′ +1 , j,t ∩ X t ∩ I j | > | S ( i ′ , j,t ∩ X t ∩ I j | .Proof. Assume that i ′ is an integer in [0 , i − such that either | Y ( i ′ , , ∩ I ◦ j ∩ X V ( T t ) − X t | 6 = | Y ( i ′ , ,s +2) ∩ I ◦ j ∩ X V ( T t ) − X t | or | Y ( i ′ , − , ∩ I ◦ j ∩ X V ( T t ) − X t | 6 = | Y ( i ′ , − ,w ) ∩ I ◦ j ∩ X V ( T t ) − X t | . Let t ′ be the node of T with σ T ( t ′ ) = i ′ . Then either W ( i ′ , − ,k,q )2 ∩ I ◦ j ∩ X V ( T t ) − ( X t ∪ Y ( i ′ , − ,k,q ) ) = ∅ for some k ∈ [0 , w − and q ∈ [0 , s + 1] , or W ( i ′ , ,k )2 ∩ I ◦ j ∩ X V ( T t ) − ( X t ∪ Y ( i ′ , ,k ) ) = ∅ for some k ∈ [0 , s + 1] . For the former, define ℓ = − ; for the latter, define ℓ = 0 . So there existsa c -monochromatic path P of color α + 1 contained in G [ W ( i ′ ,ℓ ,α )0 ] intersecting X t ′ such thateither ℓ = − and A L ( i ′ , − ,k,α ) ( V ( P )) ∩ Z t ′ ∩ I ◦ j ∩ X V ( T t ) − X t = ∅ for some k ∈ [0 , w − , or ℓ = 0 and A L ( i ′ , ,α ) ( V ( P )) ∩ Z t ′ ∩ I ◦ j ∩ X V ( T t ) − X t = ∅ . If ℓ = − , then let β ( ℓ ) := ( − , k, α ) and β ′ ( ℓ ) := ( − , k, α + 1) ; if ℓ = 0 , then let β ( ℓ ) := (0 , α ) and β ′ ( ℓ ) := (0 , α + 1) . So A L ( i ′ ,β ( ℓ ( V ( P )) ∩ Z t ′ ∩ I ◦ j ∩ X V ( T t ) − X t = ∅ and V ( P ) ⊆ W ( i ′ ,β ( ℓ ))0 .Note that t ∈ V ( T t ′ ) . So V ( P ) ∩ X t = ∅ . Since A L ( i ′ ,β ( ℓ ( V ( P )) ∩ I ◦ j = ∅ , V ( P ) ⊆ I j . Sothere exist v ∈ X V ( T t ) , u ∈ A L ( i ′ ,β ( ℓ ( { v } ) ∩ ( Y ( i ′ ,β ′ ( ℓ )) − Y ( i ′ ,β ( ℓ )) ) ∩ Z t ′ ∩ I ◦ j ∩ X V ( T t ) − X t = ∅ ,and a subpath P ′ of P contained in G [ W ( i ′ ,β ( ℓ ))0 ] from X t to v internally disjoint from X t .Let x be an end of P ′ in X t . Since Z t ′ ∩ I ◦ j ∩ X V ( T t ) − X t = ∅ , t ∈ V ( T j,t ′ ) − ∂T j,t ′ . Since V ( P ′ ) ⊆ W ( i ′ ,β ( ℓ ))0 , we know x ∈ S ( i ′ +1 , j,t .We may assume that x ∈ S ( i ′ , j,t , for otherwise we are done. So there exists i ′′ ∈ [0 , i ′ − with t ∈ V ( T j,t i ′′ ) − ∂T j,t i ′′ , where t i ′′ is the node of T with σ T ( t i ′′ ) = i ′′ , such that x ∈ S ( i ′′ +1 , j,t − S ( i ′′ , j,t . Hence there exist ℓ ′′ ∈ { , − } and k ′′ ∈ [0 , w − such that x ∈ W ( i ′′ ,ℓ ′′ ,k ′′ )0 .So there exists a monochromatic path Q in G [ Y ( i ′′ ,ℓ ′′ ,k ′′ ) ] from X t i ′′ to x internally disjointfrom X t i ′′ . Since x ∈ V ( P ′ ) ∩ Y ( i ′′ ,ℓ ′′ ,k ′′ ) , there exists a maximal subpath P ∗ of P ′ contained in G [ Y ( i ′′ ,ℓ ′′ ,k ′′ ) ] containing x . So Q ∪ P ∗ is a monochromatic connected subgraph in G [ Y ( i ′′ ,ℓ ′′ ,k ′′ ) ] intersecting X t i ′′ . Hence P ∗ ⊆ W ( i ′′ ,ℓ ′′ ,k ′′ )0 .Suppose P ∗ = P ′ . Then there exists a ∈ V ( P ∗ ) and b ∈ N P ′ ( a ) ∩ V ( P ′ ) − V ( P ∗ ) . So b Y ( i ′′ ,ℓ ′′ ,k ′′ ) by the maximality of P ∗ . Hence b ∈ Y ( i ′ ,β ( ℓ )) − Y ( i ′′ ,ℓ ′′ ,k ′′ ) . For each i ′′′ ∈ [ i ′′ , i ′ ] ,let t i ′′′ be the node of T with σ T ( t i ′′′ ) = i ′′′ . If b ∈ Z t i ′′ , then b ∈ W ( i ′′ , − ,k ′′ ,c ( a ) − ∪ W ( i ′′ , ,k ′′ )2 ,and hence c ( a ) = c ( b ) since a ∈ W ( i ′′ ,ℓ ′′ ,k ′′ )0 , a contradiction. So b Z t i ′′ . In particular, b ∈ I ◦ j .56ince b ∈ I ◦ j and b ∈ Y ( i ′ ,β ( ℓ )) , there exists a minimum i ∗ ∈ [ i ′′ , i ′ ] such that b ∈ V ( T j,t i ∗ ) . So b ∈ Z t i ∗ and i ∗ > i ′′ + 1 . Therefore, b Z t i ′′′ for every i ′′′ ∈ [ i ′′ + 1 , i ∗ − . Hence b Y ( i ∗ , − , .Since V ( P ∗ ) ⊆ Y ( i ′′ ,ℓ ′′ ,k ′′ ) ⊆ Y ( i ∗ , − , , a ∈ W ( i ∗ , − ,k ∗ )0 ∪ W ( i ∗ , ,c ( a ) − for some k ∗ ∈ [0 , w − .So b ∈ W ( i ∗ , − ,k ∗ ,c ( a ) − ∪ W ( i ∗ , ,c ( a ) − , and hence c ( b ) = c ( a ) , a contradiction.Hence P ∗ = P ′ . So P ′ = P ∗ ⊆ W ( i ′′ ,ℓ ′′ ,k ′′ )0 . Since i ′′ < i ′ and t ∈ V ( T j,t i ′′ ) − ∂T j,t i ′′ , Z t i ′ ∩ X V ( T t ) ⊆ Z t i ′′ ∩ X V ( T t ) . So u ∈ A L ( i ′ ,β ( ℓ ( V ( P ′ )) ∩ Z t i ′ ∩ X V ( T t ) ⊆ A L ( i ′′ ,ℓ ′′ ,k ′′ ) ( W ( i ′′ ,ℓ ′′ ,k ′′ )0 ) ∩ Z t i ′′ ∩ X V ( T t ) ⊆ ( W ( i ′′ , − ,k ′′′ ,α )2 ∪ W ( i ′′ , ,k ′′ )2 ) ∩ Z t i ′′ ∩ X V ( T t ) for some k ′′′ ∈ [0 , w − . Hence α + 1 L ( i ′′ ,ℓ ′′ ,k ′′ +1) ( u ) ⊆ L ( i ′ ,β ( ℓ )) ( u ) . That is, u A L ( i ′ ,β ( ℓ ( V ( P ′ )) , a contradiction. Claim 4.12.12. Let i ∈ N and let t be the node of T with σ T ( t ) = i . Let j ∈ [ |V| − . Let S := { i ′ ∈ [0 , i − 1] : | Y ( i ′ , − , ∩ I ◦ j ∩ X V ( T t ) − X t | 6 = | Y ( i ′ , ,s +2) ∩ I ◦ j ∩ X V ( T t ) − X t |} . Then | S | w .Proof. Let i ′ ∈ S . Since | Y ( i ′ , − , ∩ I ◦ j ∩ X V ( T t ) − X t | 6 = | Y ( i ′ , ,s +2) ∩ I ◦ j ∩ X V ( T t ) − X t | , we knoweither | Y ( i ′ , − , ∩ I ◦ j ∩ X V ( T t ) − X t | 6 = | Y ( i ′ , − ,w ) ∩ I ◦ j ∩ X V ( T t ) − X t | or | Y ( i ′ , , ∩ I ◦ j ∩ X V ( T t ) − X t | 6 = | Y ( i ′ , ,s +2) ∩ I ◦ j ∩ X V ( T t ) − X t | . By Claim 4.12.11, | S ( i ′ +1 , j,t ∩ X t ∩ I j | > | S ( i ′ , j,t ∩ X t ∩ I j | . Hence | S | | X t ∩ I j | w . Claim 4.12.13. Let i ∈ N and let t be the node of T with σ T ( t ) = i . Let j ∈ [ |V| − . Then | Y ( i, − , ∩ I ◦ j ∩ X V ( T t ) | η .Proof. Let S := { i ′ ∈ [0 , i − 1] : | Y ( i ′ , ,s +2) ∩ I ◦ j ∩ X V ( T t ) − X t | > | Y ( i ′ , − , ∩ I ◦ j ∩ X V ( T t ) − X t |} .By Claim 4.12.12, | S | w . Let S := { i ′ ∈ [0 , i − 1] : | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ I ◦ j ∩ X V ( T t ) − X t | > | Y ( i ′ , ,s +2) ∩ I ◦ j ∩ X V ( T t ) − X t |} . Since I ◦ j ⊆ I j , | S | w by Claim 4.12.6.For every i ′ ∈ [0 , i ] , let t i ′ be the node of T with σ T ( t i ′ ) = i ′ .For every i ′ ∈ S , ( Y ( i ′ , ,s +2) − Y ( i ′ , − , ) ∩ I ◦ j ∩ X V ( T t ) − X t ⊆ X V ( T j,ti ′ ) ∩ I j ∩ X V ( T t ) − X t . Sofor every i ′ ∈ S , | ( Y ( i ′ , ,s +2) − Y ( i ′ , − , ) ∩ I ◦ j ∩ X V ( T t ) − X t | | Y ( i ′ , ,s +2) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,ti ′ ) | g ( s + 2) by Statement 2 in Claim 4.12.10.For every i ′ ∈ S , ( Y ( i ′ , | V ( T ) | +1 ,s +2) − Y ( i ′ , ,s +2) ) ∩ I ◦ j ∩ X V ( T t ) − X t ⊆ X V ( T j,ti ′ ) ∩ I j ∩ X V ( T t ) − X t .So for every i ′ ∈ S , | ( Y ( i ′ , | V ( T ) | +1 ,s +2) − Y ( i ′ , ,s +2) ) ∩ I ◦ j ∩ X V ( T t ) − X t | | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ I j ∩ X V ( T t ) ∩ X V ( T j,ti ′ ) | g (4 w ) by Statement 4 in Claim 4.12.10.Note that for every i ′ ∈ [0 , i − , ( Y ( i ′ +1 , − , − Y ( i ′ , | V ( T ) | +1 ,s +2) ) ∩ X V ( T t ) ⊆ X t . Hence | Y ( i, − , ∩ I ◦ j ∩ X V ( T t ) − X t | | Y (0 , − , ∩ I ◦ j ∩ X V ( T t ) − X t | + | S | · g ( s + 2) + | S | · g (4 w ) w g ( s + 2) + 3 w g (4 w ) . Therefore, | Y ( i, − , ∩ I ◦ j ∩ X V ( T t ) | w g ( s + 2) + 3 w g (4 w ) + | X t ∩ I j | w g ( s + 2) + 3 w g (4 w ) + w = η . Claim 4.12.14. Let i ∈ N , and let t be the node of T with σ T ( t ) = i . Let j ∈ [ | V | − .Let Z j be the set obtained from the j -th belt by deleting I j − , ∪ I j, , where I , = ∅ . Let S := { i ′ ∈ [0 , i − 1] : | Y ( i ′ , − , ∩ X V ( T t ) ∩ Z j − X t | < | Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ Z j − X t |} . Then | S | w .Proof. Let Z j := Z j ∪ I j − ∪ I j . For any i ′ ∈ N and k ∈ [0 , s + 2] , denote U ( i ′ ,k ) by U ( i ′ , ,k ) and let t i ′ be the node of T with σ T ( t i ′ ) = i ′ . We shall show that for every i ′ ∈ S , | U ( i ′ , − , ∩ X t ∩ Z j | < | U ( i ′ , ,s +2) ∩ X t ∩ Z j | . 57uppose to the contrary that there exists i ′ ∈ S such that | U ( i ′ , − , ∩ X t ∩ Z j | > | U ( i ′ , ,s +2) ∩ X t ∩ Z j | . Since U ( i ′ , − , ⊆ U ( i ′ , ,s +2) , U ( i ′ , − , ∩ X t ∩ Z j = U ( i ′ , ,s +2) ∩ X t ∩ Z j . Since | Y ( i ′ , − , ∩ X V ( T t ) ∩ Z j − X t | < | Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ Z j − X t | , there exist ℓ ∈ {− , } and k ∈ [0 , w − such that W ( i ′ ,ℓ,k )0 = ∅ and A L ( i ′ ,ℓ,k ) ( W ( i ′ ,ℓ,k )0 ) ∩ Z j ∩ X V ( T t ) − X t = ∅ . Since i ′ < i ,there exists a monochromatic path P in G [ Y ( i ′ ,ℓ,k ) ] from X t i ′ to X V ( T t ) intersecting N G [ Z j ] ,and A L ( i ′ ,ℓ,k ) ( V ( P )) ∩ ( Y ( i ′ , ,s +2) − Y ( i ′ , − , ) ∩ Z j = ∅ . Since P is a monochromatic path, V ( P ) ⊆ Z j . Let x ∈ V ( P ) ∩ X t ∩ Z j . So x ∈ U ( i ′ , ,s +2) ∩ X t ∩ Z j = U ( i ′ , − , ∩ X t ∩ Z j . Hencethere exist i ′′ ∈ [0 , i ′ − , ℓ ′′ ∈ {− , } and k ′′ ∈ [0 , w − such that x ∈ W ( i ′′ ,ℓ ′′ ,k ′′ )0 . So thereexists a monochromatic path P ′′ contained in G [ Y ( i ′′ ,ℓ ′′ ,k ′′ ) ] from X t i ′′ to x . Hence P ∪ P ′′ is amonochromatic connected subgraph in G [ Y ( i ′ ,ℓ,k ) ] intersecting x and X t i ′′ . Let Q be a maximalconnected subgraph of P ∪ P ′′ contained in G [ Y ( i ′′ ,ℓ ′′ ,k ′′ ) ] containing x and intersecting X t i ′′ .So Q contains P ′′ , and V ( Q ) ⊆ W ( i ′′ ,ℓ ′′ ,k ′′ )0 . If V ( P ) ⊆ V ( Q ) , then A L ( i ′′ ,ℓ ′′ ,k ′′ ) ( V ( P )) ∩ Z j ⊆ Y ( i ′′ +1 , − , ⊆ Y ( i ′ , − , , so A L ( i ′ ,ℓ,k ) ( V ( P )) ∩ ( Y ( i ′ , ,s +2) − Y ( i ′ , − , ) ∩ Z j = ∅ , a contradiction.So V ( P ) V ( Q ) . Hence there exist u ∈ V ( Q ) and v ∈ N G ( u ) ∩ V ( P ∪ P ′′ ) − V ( Q ) such that c ( u ) = c ( v ) . But c ( v ) = c ( u ) L ( i ′′ ,ℓ ′′ ,k ′′ +1) ( v ) by the definition of ( Y ( i ′′ ,ℓ ′′ ,k ′′ +1) , L ( i ′′ ,ℓ ′′ ,k ′′ +1) ) ,a contradiction.Hence | U ( i ′ , − , ∩ X t ∩ Z j | < | U ( i ′ , ,s +2) ∩ X t ∩ Z j | for every i ′ ∈ S . Therefore, | S | | X t ∩ Z j | w . Claim 4.12.15. Let i, i ′ ∈ N with i ′ < i , and let t be a node of T with σ T ( t ) = i . Let j ∈ [ | V | − . Let Z j be the set obtained from the j -th belt by deleting I j − , ∪ I j, , where I , = ∅ . Then | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ Z j ∩ X V ( T t ) | g ( | Y ( i ′ , − , ∩ Z j ∩ X V ( T t ) | ) .Proof. Note that N G [ Z j ] ⊆ Z j ∪ I ◦ j − ∪ I ◦ j . For every k ∈ [0 , w − and q ∈ [0 , s + 1] , | ( Y ( i ′ , − ,k,q +1) − Y ( i ′ , − ,k,q ) ) ∩ Z j ∩ X V ( T t ) − X t | | A L ( i ′ , − ,k,q ) ( W ( i ′ , − ,k,q )1 ) ∩ Z j ∩ X V ( T t ) − X t | | A L ( i ′ , − ,k,q ) ( W ( i ′ , − ,k,q )1 ∩ ( Z j ∪ I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) ) ∩ Z j ∩ X V ( T t ) − X t | | N > sG ( W ( i ′ , − ,k,q )1 ∩ ( Z j ∪ I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) ) | | N > sG ( Y ( i ′ , − ,k ) ∩ ( Z j ∪ I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) ) | f ( | Y ( i ′ , − ,k ) ∩ ( Z j ∪ I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) | ) f ( | Y ( i ′ , − ,k ) ∩ Z j ∩ X V ( T t ) | + | Y ( i ′ , − ,k ) ∩ ( I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) | ) f ( | Y ( i ′ , − ,k ) ∩ Z j ∩ X V ( T t ) | + | Y ( i, − , ∩ ( I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) | ) f ( | Y ( i ′ , − ,k ) ∩ Z j ∩ X V ( T t ) | + 2 η ) , where the last inequality follows from Claim 4.12.13. So for every k ∈ [0 , w − , | Y ( i ′ , − ,k +1) ∩ Z j ∩ X V ( T t ) | | X t ∩ Z j | + | Y ( i ′ , − ,k,s +2) ∩ Z j ∩ X V ( T t ) − X t | | X t ∩ Z j | + | Y ( i ′ , − ,k, ∩ Z j ∩ X V ( T t ) − X t | + ( s + 2) · f ( | Y ( i ′ , − ,k ) ∩ Z j ∩ X V ( T t ) | + 2 η ) w + | Y ( i ′ , − ,k ) ∩ Z j ∩ X V ( T t ) − X t | + ( s + 2) · f ( | Y ( i ′ , − ,k ) ∩ Z j ∩ X V ( T t ) | + 2 η ) ( s + 2) · f ( | Y ( i ′ , − ,k ) ∩ Z j ∩ X V ( T t ) | + 2 η ) . Hence it is easy to verify that for every k ∈ [0 , w ] , | Y ( i ′ , − ,k ) ∩ Z j ∩ X V ( T t ) | g ( k, | Y ( i ′ , − , ∩ Z j ∩ X V ( T t ) | ) by induction on k . Therefore | Y ( i ′ , , ∩ Z j ∩ X V ( T t ) | = | Y ( i ′ , − ,w ) ∩ Z j ∩ X V ( T t ) | g ( w , | Y ( i ′ , − , ∩ Z j ∩ X V ( T t ) | ) .For every k ∈ [0 , s + 1] , | ( Y ( i ′ , ,k +1) − Y ( i ′ , ,k ) ) ∩ Z j ∩ X V ( T t ) − X t | | A L ( i ′ , ,k ) ( W ( i ′ , ,k )0 ) ∩ Z j ∩ X V ( T t ) − X t | | A L ( i ′ , ,k ) ( W ( i ′ , ,k )0 ∩ ( Z j ∪ I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) ) ∩ Z j ∩ X V ( T t ) − X t | | N > sG ( W ( i ′ , ,k )0 ∩ ( Z j ∪ I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) ) | | N > sG ( Y ( i ′ , ,k ) ∩ ( Z j ∪ I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) ) | f ( | Y ( i ′ , ,k ) ∩ ( Z j ∪ I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) | ) f ( | Y ( i ′ , ,k ) ∩ Z j ∩ X V ( T t ) | + | Y ( i ′ , ,k ) ∩ ( I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) | ) f ( | Y ( i ′ , ,k ) ∩ Z j ∩ X V ( T t ) | + | Y ( i, − , ∩ ( I ◦ j − ∪ I ◦ j ) ∩ X V ( T t ) | ) f ( | Y ( i ′ , ,k ) ∩ Z j ∩ X V ( T t ) | + 2 η ) , where the last inequality follows from Claim 4.12.13. So for every k ∈ [0 , s + 1] , | Y ( i ′ , ,k +1) ∩ Z j ∩ X V ( T t ) | | X t ∩ Z j | + | Y ( i ′ , ,k +1) ∩ Z j ∩ X V ( T t ) − X t | | X t ∩ Z j | + | Y ( i ′ , ,k ) ∩ Z j ∩ X V ( T t ) − X t | + f ( | Y ( i ′ , ,k ) ∩ Z j ∩ X V ( T t ) | + 2 η ) f ( | Y ( i ′ , ,k ) ∩ Z j ∩ X V ( T t ) | + 2 η ) . Therefore, | Y ( i ′ , ,s +2) ∩ Z j ∩ X V ( T t ) | g ( s + 2 , | Y ( i ′ , , ∩ Z j ∩ X V ( T t ) | ) .Note that ( Y ( i ′ , | V ( T ) | +1 ,s +2) − Y ( i ′ , ,s +2) ) ∩ Z j ∩ X V ( T t ) ⊆ ( Y ( i ′ , | V ( T ) | +1 ,s +2) − Y ( i ′ , ,s +2) ) ∩ ( I j − ∪ I j ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) ⊆ Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ ( I j − ∪ I j ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) , where t ′ is the node of T with σ T ( t ′ ) = i ′ . So | ( Y ( i ′ , | V ( T ) | +1 ,s +2) − Y ( i ′ , ,s +2) ) ∩ Z j ∩ X V ( T t ) | | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ ( I j − ∪ I j ) ∩ X V ( T j,t ′ ) ∩ X V ( T t ) | g (4 w ) by Statement 4 in Claim 4.12.10.Therefore, | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ Z j ∩ X V ( T t ) | | Y ( i ′ , ,s +2) ∩ Z j ∩ X V ( T t ) | + 2 g (4 w ) g ( s + 2 , | Y ( i ′ , , ∩ Z j ∩ X V ( T t ) | ) + 2 g (4 w ) g ( s + 2 , g ( w , | Y ( i ′ , − , ∩ Z j ∩ X V ( T t ) | )) + 2 g (4 w )= g ( | Y ( i ′ , − , ∩ Z j ∩ X V ( T t ) | ) . laim 4.12.16. Let i ∈ N , and let t be a node of T with σ T ( t ) = i . Let j ∈ [ | V | − .Let Z j be the set obtained from the j -th belt by deleting I j − , ∪ I j, , where I , = ∅ . Then | Y ( i, − , ∩ Z j ∩ X V ( T t ) | η .Proof. Let S := { i ′ ∈ [0 , i − 1] : | Y ( i ′ , − , ∩ X V ( T t ) ∩ Z j − X t | < | Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ Z j − X t |} . By Claim 4.12.14, | S | w . Let S := { i ′ ∈ [0 , i − 1] : | Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ Z j − X t | < | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ Z j − X t |} . Note that for every i ′ ∈ S , ( Y ( i ′ , | V ( T ) | +1 ,s +2) − Y ( i ′ , ,s +2) ) ∩ X V ( T t ) ∩ Z j − X t ⊆ ( Y ( i ′ , | V ( T ) | +1 ,s +2) − Y ( i ′ , ,s +2) ) ∩ X V ( T t ) ∩ Z j ∩ ( I j − ∪ I j ) − X t , where I = ∅ . So | S | |{ i ′ ∈ [0 , i − 1] : | Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − − X t | < | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ I j − − X t |}| + |{ i ′ ∈ [0 , i − 1] : | Y ( i ′ , ,s +2) ∩ X V ( T t ) ∩ I j − X t | < | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ I j − X t |}| w by Claim 4.12.6. Let S := { i ′ ∈ [0 , i − 1] : | Y ( i ′ , − , ∩ X V ( T t ) ∩ Z j − X t | < | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) ∩ Z j − X t |} . So | S | | S | + | S | w . Let S := { i ′ ∈ [0 , i − 1] : | Y ( i ′ , − , ∩ X V ( T t ) ∩ Z j − X t | < | Y ( i ′ +1 , − , ∩ X V ( T t ) ∩ Z j − X t |} . Since for every i ′ ∈ [0 , i − , Y ( i ′ +1 , − , ∩ Z j ∩ X V ( T t ) ⊆ ( Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ Z j ∩ X V ( T t ) ) ∪ ( X t ∩ Z j ) ,we know | S | | S | + | X t ∩ Z j | w .By Claim 4.12.15, for every i ′ ∈ S , | Y ( i ′ +1 , − , ∩ Z j ∩ X V ( T t ) | | Y ( i ′ , | V ( T ) | +1 ,s +2) ∩ Z j ∩ X V ( T t ) | + | X t ∩ Z j | g ( | Y ( i ′ , − , ∩ Z j ∩ X V ( T t ) | ) + | X t ∩ Z j | g ( | Y ( i ′ , − , ∩ Z j ∩ X V ( T t ) | ) + w .Note that | Y (0 , − , ∩ Z j ∩ X V ( T t ) | | X t ∩ Z j | w = g (0) . Then it is easy to verifythat | Y ( i, − , ∩ Z j ∩ X V ( T t ) | g ( | S | ) by induction on the elements in S . Since | S | w , | Y ( i, − , ∩ Z j ∩ X V ( T t ) | g (8 w ) = η . Claim 4.12.17. Let i ∈ N and let t ∈ V ( T ) with σ T ( t ) = i . Let p be the parent of t if t = r ∗ ;otherwise let F j,p be the set mentioned in the algorithm when t = r ∗ . Then • for every q ∈ F j,p , Y ( i, − , ∩ I j ∩ X V ( T q ) − X q = ∅ , and • for every q ∈ F ′ j,t , Y ( i +1 , − , ∩ I j ∩ X V ( T q ) − X q = ∅ .Proof. We shall prove this claim by induction on i . We first assume i = 0 . So t = r ∗ . Since F j, = ∅ , T is the only F j, ∩ V ( T t ) -part containing t , and ∂T = ∅ . Since Y (0 , − , = ∅ , weknow F j,T = ∅ . Hence the set F j,p defined in the algorithm is ∅ . So Statement 1 in this claimholds. Hence we may assume that q ∈ F ′ j,t . Since F j,p = ∅ , q ∈ F ′ j,T ′ for some ∅ -part T ′ of T t containing t . So T ′ = T , and F ′ j,T ′ is a ( T, X | X V ( T ) ∩ I j , Y (1 , − , ∩ I j ) -fence. By the definition ofa fence (Statement 2 in Lemma 4.9), there exist at least two F ′ j,T ′ ∩ V ( T q ) -parts T ′′ satisfying60 ∈ ∂T ′′ and X V ( T ′′ ) ∩ Y (1 , − , ∩ I j − X q = ∅ . Note that some such part T ′′ is contained in T q . So Y (1 , − , ∩ I j ∩ X V ( T q ) − X q = ∅ . Hence the claim holds when i = 0 .So we may assume i > , and the lemma holds for every smaller i . We first assume q ∈ F j,p and prove Statement 1. So either q ∈ F ′ j,p or q ∈ F j,T ′ − F ′ j,p for some F ′ j,p ∩ V ( T t ) -part T ′ of T t containing t . If F ′ j,p ∩ V ( T q ) = ∅ , then there exists q ′ ∈ F ′ j,p ∩ V ( T q ) and Y ( i, − , ∩ I j ∩ X V ( T q ) − X q ⊇ Y ( i p +1 , − , ∩ I j ∩ X V ( T q ′ ) − X q ′ = ∅ by the induction hypothesis,where i p = σ T ( p ) . So we may assume F ′ j,p ∩ V ( T q ) = ∅ . In particular, q ∈ F j,T ′ − F ′ j,p for some F ′ j,p ∩ V ( T t ) -part T ′ of T t containing t , and ∂T ′ ∩ V ( T q ) = ∅ . By the definitionof a fence (Statement 2 in Lemma 4.9), there exist at least two F j,T ′ -parts T ′′ satisfying q ∈ ∂T ′′ and ( Y ( i, − , ∪ X ∂T ′ ) ∩ X V ( T ′′ ) ∩ I j − X q = ∅ . So one such T ′′ is contained in T q . Since ∂T ′ ∩ V ( T q ) = ∅ , we have Y ( i, − , ∩ I j ∩ X V ( T q ) − X q ⊇ ( Y ( i, − , ∪ X ∂T ′ ) ∩ X V ( T ′′ ) ∩ I j ∩− X q = ∅ .This proves Statement 1 of this claim.Similarly, when q ∈ F ′ j,t ′ , Y ( i +1 , − , ∩ I j ∩ X V ( T q ) − X q = ∅ and Statement 2 holds. Claim 4.12.18. Let t ∈ V ( T ) . Let j ∈ [ |V| − . Then | X ∂T j,t ∩ I j | η .Proof. Let p be the parent of t if t = r ∗ ; otherwise, let F j,p be the set mentioned in thealgorithm when t = r ∗ . Note that ∂T j,t ⊆ F j,p . By Claim 4.12.17, for every q ∈ ∂T j,t , Y ( i, − , ∩ I j ∩ X V ( T q ) − X q = ∅ . Since X V ( T q ′ ) − X q ′ is disjoint from X V ( T q ′′ ) − X q ′′ for anydistinct q ′ , q ′′ of ∂T j,t , we have | ∂T j,t | | Y ( i, − , ∩ I j | . Hence | ∂T j,t | | Y ( i, − , ∩ I j ∩ X V ( T t ) − X t | | Y ( i, − , ∩ I j ∩ X V ( T t ) | | Y ( i, − , ∩ ( Z j ∪ Z j +1 ∪ I ◦ j ) ∩ X V ( T t ) | η + 2 η by Claims 4.12.13 and 4.12.16, where Z j and Z j +1 are the sets obtained from the j -th beltby deleting I j − , ∪ I j, and from the ( j + 1) -th belt by deleting I j, ∪ I j +1 , , respectively.Therefore, | X ∂T j,t ∩ I j | | ∂T j,t | · w ( η + 2 η ) w = η . Claim 4.12.19. Let i ∈ N and let t ∈ V ( T ) with σ T ( t ) = i . Let j ∈ [ |V| − . Then | Y ( i, , ∩ I j ∩ X V ( T t ) | g ( w ) .Proof. Note that for every k ∈ [0 , w − , W ( i, − ,k )0 ∩ X V ( T t ) ⊆ S |V|− j ′ =1 S S ∈ S ◦ j ′ S . So ( Y ( i, − ,k +1) − Y ( i, − ,k ) ) ∩ X V ( T t ) ⊆ N G [ W ( i, − ,k )0 ] ∩ X V ( T t ) ⊆ N G [ |V|− [ j ′ =1 [ S ∈ S ◦ j ′ S ] ⊆ |V|− [ j ′ =1 I ◦ j ′ ⊆ |V|− [ j ′ =1 I j ′ . So for every k ∈ [0 , w − and q ∈ [0 , s + 1] , ( Y ( i, − ,k,q +1) − Y ( i, − ,k,q ) ) ∩ I j ∩ X V ( T t ) − X t ⊆ A L ( i, − ,k, ( Y ( i, − ,k,q )1 ∩ I ◦ j ) ∩ I j ∩ X V ( T t ) − X t ⊆ N > sG ( Y ( i, − ,k,q ) ∩ I ◦ j ) ∩ I j ∩ X V ( T t ) − X t ⊆ N > sG ( Y ( i, − ,k,q ) ∩ I ◦ j ∩ X V ( T t ) ) . Hence for every k ∈ [0 , w − and q ∈ [0 , s + 1] , | ( Y ( i, − ,k,q +1) − Y ( i, − ,k,q ) ) ∩ I j ∩ X V ( T t ) − X t | | N > sG ( Y ( i, − ,k,q ) ∩ I ◦ j ∩ X V ( T t ) ) | f ( | Y ( i, − ,k,q ) ∩ I j ∩ X V ( T t ) | ) . Therefore, for every k ∈ [0 , w − and q ∈ [0 , s + 1] , | ( Y ( i, − ,k,q +1) − Y ( i, − ,k,q ) ) ∩ I j ∩ X V ( T t ) | | ( Y ( i, − ,k,q +1) − Y ( i, − ,k,q ) ) ∩ I j ∩ X V ( T t ) − X t | + | X t ∩ I j | f ( | Y ( i, − ,k,q ) ∩ I j ∩ X V ( T t ) | ) + w . | Y ( i, − ,k,q +1) ∩ I j ∩ X V ( T t ) | | ( Y ( i, − ,k,q +1) − Y ( i, − ,k,q ) ) ∩ I j ∩ X V ( T t ) | + | Y ( i, − ,k,q ) ∩ I j ∩ X V ( T t ) | f ( | Y ( i, − ,k,q ) ∩ I j ∩ X V ( T t ) | ) + w + | Y ( i, − ,k,q ) ∩ I j ∩ X V ( T t ) | = f ( | Y ( i, − ,k,q ) ∩ I j ∩ X V ( T t ) | ) + w . Note that for every k ∈ [0 , w − , | Y ( i, − ,k, ∩ I j ∩ X V ( T t ) | g (0 , | Y ( i, − ,k, ∩ I j ∩ X V ( T t ) | ) .So it is easy to verify that for every k ∈ [0 , w − , for every q ∈ [0 , s + 2] , | Y ( i, − ,k,q ) ∩ I j ∩ X V ( T t ) | g ( q, | Y ( i, − ,k, ∩ I j ∩ X V ( T t ) | ) by induction on q . That is, for every k ∈ [0 , w − , | Y ( i, − ,k +1) ∩ I j ∩ X V ( T t ) | = | Y ( i, − ,k,s +2) ∩ I j ∩ X V ( T t ) | g ( s + 2 , | Y ( i, − ,k, ∩ I j ∩ X V ( T t ) | ) = g ( s + 2 , | Y ( i, − ,k ) ∩ I j ∩ X V ( T t ) | ) .Note that | Y ( i, − , ∩ I j ∩ X V ( T t ) | η + 2 η = g (0) by Claims 4.12.13 and 4.12.16. So itis easy to verify that for every k ∈ [0 , w ] , | Y ( i, − ,k ) ∩ I j ∩ X V ( T t ) | g ( k ) by induction on k .Therefore, | Y ( i, , ∩ I j ∩ X V ( T t ) | = | Y ( i, − ,w ) ∩ I j ∩ X V ( T t ) | g ( w ) . Claim 4.12.20. Let i ∈ N and let t ∈ V ( T ) with σ T ( t ) = i . Let j ∈ [ |V| − . Then | Y ( i, ,s +2) ∩ B j ∩ X V ( T t ) | η , where B j is the j -th belt.Proof. Let a j and b j be the integers such that B j = S b j α = a j V α . For every k ∈ [0 , s + 2] , let R k := S b j +( s +2) − kα = a j − ( s +2)+ k V α . Let Z j := B j − ( I j − , ∪ I j, ) , where I , = ∅ . Note that B j = R s +2 ⊆ R k +1 ⊆ I ◦ j − ∪ Z j ∪ I ◦ j = R for every k ∈ [0 , s + 1] .Note that for every k ∈ [0 , s + 1] , ( Y ( i, ,k +1) − Y ( i, ,k ) ) ∩ R k +1 ∩ X V ( T t ) − X t ⊆ A L ( i, ,k ) ( W ( i, ,k )0 ) ∩ R k +1 ∩ X V ( T t ) − X t ⊆ A L ( i, ,k ) ( W ( i, ,k )0 ∩ R k ) ∩ R k +1 ∩ X V ( T t ) − X t ⊆ A L ( i, ,k ) ( Y ( i, ,k ) ∩ R k ) ∩ R k +1 ∩ X V ( T t ) − X t ⊆ N > sG ( Y ( i, ,k ) ∩ R k ∩ X V ( T t ) ) . So for every k ∈ [0 , s + 1] , | ( Y ( i, ,k +1) − Y ( i, ,k ) ) ∩ R k +1 ∩ X V ( T t ) − X t | | N > sG ( Y ( i, ,k ) ∩ R k ∩ X V ( T t ) ) | f ( | Y ( i, ,k ) ∩ R k ∩ X V ( T t ) | ) . Hence for every k ∈ [0 , s + 1] , | Y ( i, ,k +1) ∩ R k ∩ X V ( T t ) | | Y ( i, ,k ) ∩ R k ∩ X V ( T t ) | + | ( Y ( i, ,k +1) − Y ( i, ,k ) ) ∩ R k +1 ∩ X V ( T t ) − X t | + | X t ∩ R k +1 | | Y ( i, ,k ) ∩ R k ∩ X V ( T t ) | + f ( | Y ( i, ,k ) ∩ R k ∩ X V ( T t ) | ) + w = f ( | Y ( i, ,k ) ∩ R k ∩ X V ( T t ) | ) + w . Recall that ( Y ( i, , − Y ( i, − , ) ∩ B j ∩ X V ( T t ) ⊆ N G [ S S ∈S ◦ j − ∪S ◦ j S ] ∩ X V ( T t ) ⊆ ( I j − ∪ I j ) ∩ X V ( T t ) . So | Y ( i, , ∩ R ∩ X V ( T t ) | = | Y ( i, , ∩ ( I ◦ j − ∪ Z j ∪ I ◦ j ) ∩ X V ( T t ) | | Y ( i, , ∩ I j − ∩ X V ( T t ) | + | Y ( i, , ∩ ( Z j − ( I j − ∪ I j )) ∩ X V ( T t ) | + | Y ( i, , ∩ I j ∩ X V ( T t ) | g ( w ) + | Y ( i, − , ∩ Z j ∩ X V ( T t ) | + g ( w ) g ( w ) + η g (0) by Claims 4.12.16 and 4.12.19. Hence it is easy to verify that for every k ∈ [0 , s + 2] , | Y ( i, ,k ) ∩ R k ∩ X V ( T t ) | g ( k ) . In particular, | Y ( i, ,s +2) ∩ B j ∩ X V ( T t ) | = | Y ( i, ,s +2) ∩ R s +2 ∩ X V ( T t ) | g ( s + 2) = η Claim 4.12.21. Let M be a c -monochromatic component. If V ( M ) ∩ S |V|− j =1 I ◦ j = ∅ , then | V ( M ) | η .Proof. Let t be the node of T with minimum σ T ( t ) such that V ( M ) ∩ X V ( T t ) = ∅ . Let i = σ T ( t ) . By the minimality of i , V ( M ) ⊆ X V ( T t ) and V ( M ) ∩ X t = ∅ . We claim that V ( M ) ⊆ Y ( i, ,s +2) ∩ X V ( T t ) .Suppose to the contrary that V ( M ) Y ( i, ,s +2) . Let k ∈ [0 , s +1] such that c ( v ) = k +1 forevery v ∈ V ( M ) . Since X t ⊆ Y ( i, − , , V ( M ) ∩ Y ( i, − , ∩ X t = ∅ . So V ( M ) ∩ Y ( i, ,k ) ∩ X t = ∅ .Let M ′ be the union of all components of M [ Y ( i, ,k ) ] intersecting X t . Since V ( M ) ⊆ X V ( T t ) , V ( M ′ ) ⊆ W ( i, ,k )0 . Since V ( M ) Y ( i, ,s +2) , V ( M ) Y ( i, ,k ) , so there exists v ∈ V ( M ) − V ( M ′ ) adjacent in G to V ( M ′ ) . So v ∈ A L ( i, ,k ) ( V ( M ′ )) . Since v ∈ V ( M ) , and V ( M ) isdisjoint from S |V|− j =1 I ◦ j , v ∈ Z t . So v ∈ W ( i, ,k )2 . Since ( Y ( i, ,k +1) , L ( i, ,k +1) ) is a ( W ( i, ,k )2 , k + 1) -progress of ( Y ( i, ,k ) , L ( i, ,k ) ) , k + 1 L ( i, ,k +1) ( v ) . But c ( v ) = k + 1 , a contradiction.Therefore, V ( M ) ⊆ Y ( i, ,s +2) ∩ X V ( T t ) . Since M is a monochromatic component disjointfrom S |V|− j =1 I ◦ j , there exists j ∗ ∈ [ |V| − such that V ( M ) ⊆ Z j ∗ , where Z j ∗ is the set obtainedfrom the j ∗ -th belt by deleting I j ∗ − , ∪ I j ∗ , . So | V ( M ) | | Y ( i, ,s +2) ∩ X V ( T t ) ∩ Z j ∗ | η byClaim 4.12.20. Claim 4.12.22. Let i ∈ N and let t ∈ V ( T ) with σ T ( t ) = i . Let j ∈ [ |V| − and ℓ ∈ [0 , | V ( T ) | ] . Then | Y ( i,ℓ +1 ,s +2) ∩ I j ∩ X V ( T t ) | f s +2 ( | Y ( i,ℓ +1 , ∩ I j ∩ X V ( T t ) | + 2 η ) − η .Proof. Let B j be the j -th belt. Note that | Y ( i, | V ( T ) | +1 ,s +2) ∩ ( B j − ( I j − ∪ I j )) ∩ X V ( T t ) | = | Y ( i, ,s +2) ∩ ( B j − ( I j − ∪ I j )) ∩ X V ( T t ) | η by Claim 4.12.20. So for every ℓ ∈ [0 , | V ( T ) | ] and k ∈ [0 , s + 1] , | ( Y ( i,ℓ +1 ,k +1) − Y ( i,ℓ +1 ,k ) ) ∩ I j ∩ X V ( T t ) | | N > sG ( Y ( i,ℓ +1 ,k ) ∩ ( I j ∪ ( B j − ( I j − ∪ I j )) ∪ ( B j +1 − ( I j ∪ I j +1 ))) ∩ X V ( T t ) ) | f ( | Y ( i,ℓ +1 ,k ) ∩ I j ∩ X V ( T t ) | + 2 η ) . Hencefor every ℓ ∈ [0 , | V ( T ) | ] and k ∈ [0 , s + 1] , | Y ( i,ℓ +1 ,k +1) ∩ I j ∩ X V ( T t ) | f ( | Y ( i,ℓ +1 ,k ) ∩ I j ∩ X V ( T t ) | + 2 η ) − η . Then it is easy to show that for every ℓ ∈ [0 , | V ( T ) | ] and k ∈ [0 , s + 1] , | Y ( i,ℓ +1 ,k ) ∩ I j ∩ X V ( T t ) | f k ( | Y ( i,ℓ +1 , ∩ I j ∩ X V ( T t ) | + 2 η ) − η by induction on k . Therefore, | Y ( i,ℓ +1 ,s +2) ∩ I j ∩ X V ( T t ) | f s +2 ( | Y ( i,ℓ +1 , ∩ I j ∩ X V ( T t ) | + 2 η ) − η . Claim 4.12.23. Let i ∈ N and let t ∈ V ( T ) with σ T ( t ) = i . Let j ∈ [ |V| − . Then | Y ( i, | V ( T ) | +1 ,s +2) ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | η / , where B j ′ is the j ′ -th belt forevery j ′ ∈ [ |V| ] .Proof. By Claim 4.12.20, | Y ( i, ,s +2) ∩ ( B j ∪ B j +1 ) ∩ X V ( T t ) | η = g (0) .For every ℓ ∈ [0 , | V ( T ) | ] , by Claim 4.12.18, | Y ( i,ℓ +1 , ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | | Y ( i,ℓ,s +2) ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | + | X ∂T j,t ∩ I j | | Y ( i,ℓ,s +2) ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 ) ∩ X V ( T t ) | + η . ℓ ∈ [0 , | V ( T ) | ] , ( Y ( i,ℓ +1 ,s +2) − Y ( i,ℓ +1 , ) ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) ⊆ ( Y ( i,ℓ +1 ,s +2) − Y ( i,ℓ +1 , ) ∩ I j ∩ X V ( T t ) . So for every ℓ ∈ [0 , | V ( T ) | ] , by Claim 4.12.22, | Y ( i,ℓ +1 ,s +2) ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | | Y ( i,ℓ +1 , ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | + | Y ( i,ℓ +1 ,s +2) ∩ I j ∩ X V ( T t ) | | Y ( i,ℓ +1 , ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | + f s +2 ( | Y ( i,ℓ +1 , ∩ I j ∩ X V ( T t ) | + 2 η ) − η f s +3 ( | Y ( i,ℓ +1 , ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | + 2 η ) − η f s +3 ( | Y ( i,ℓ,s +2) ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | + η + 2 η ) . By Claim 4.12.18, | X ∂T j,t ∩ I j | η , so by Claim 4.12.2, there are at most η numbers ℓ ∈ [0 , | V ( T ) | ] such that Y ( i,ℓ +1 , ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) = Y ( i,ℓ, ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) . Hence it is straight forward to verify by induction on ℓ that | Y ( i, | V ( G ) | +1 ,s +2) ∩ (( B j ∪ B j +1 ) − ( I j − ∪ I j +1 )) ∩ X V ( T t ) | g ( η ) = η / . Claim 4.12.24. Let i ∈ N and let t ∈ V ( T ) with σ T ( t ) = i . Let j ∈ [ |V| − . Then | Y ( i, | V ( T ) | +1 ,s +2) ∩ ( B j ∪ B j +1 ) ∩ X V ( T t ) | η , where B j ′ is the j ′ -th belt for every j ′ ∈ [ |V| ] .Proof. By Claim 4.12.23, | Y ( i, | V ( T ) | +1 ,s +2) ∩ ( B j ∪ B j +1 ) ∩ X V ( T t ) | P j ′ +1 j = j ′ − | Y ( i, | V ( T ) | +1 ,s +2) ∩ (( B j ′ ∪ B j ′ +1 ) − ( I j ′ − ∪ I j ′ +1 )) ∩ X V ( T t ) | · η / η .Given any c -monochromatic component M , there uniquely exists a node r M of T suchthat V ( M ) ∩ X r M = ∅ and V ( M ) ⊆ X V ( T rM ) , and we define i M := σ T ( r M ) . Recall thatevery monochromatic component is contained in some s -segment. For every c -monochromaticcomponent M , let S M be the s -segment containing V ( M ) whose level equals the color of M . Claim 4.12.25. Let j ∈ [ |V| − , and let M be a c -monochromatic component such that S M ∩ I ◦ j = ∅ . Let i ∈ N and t be a node of T with σ T ( t ) = i such that V ( M ) ∩ X t ∗ = ∅ for some witness t ∗ ∈ ∂T j,t ∪ { t } for X t ∗ ∩ I j ⊆ W ( i,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] . Then V ( M ) ∩ X V ( T j,t ) ⊆ Y ( i, | V ( T ) | +1 ,s +2) and A L ( i, | V ( T ) | +1 ,s +2) ( V ( M ) ∩ X V ( T j,t ) ) ∩ X V ( T j,t ) = ∅ .Proof. We may assume that there exists no t ′ ∈ V ( T ) with t ∈ V ( T t ′ ) − { t ′ } and V ( T j,t ′ ) ∩ ∂T j,t = ∅ such that V ( M ) ∩ X t ′′ = ∅ for some witness t ′′ ∈ ∂T j,t ′ ∪ { t ′ } for X t ′′ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 for some ℓ ′ ∈ [ − , | V ( T ) | ] , where i ′ is the integer σ T ( t ′ ) , for otherwise we may replace t by t ′ due to the facts that T j,t ⊆ T j,t ′ and Y ( i ′ , | V ( T ) | +1 ,s +2) ⊆ Y ( i, | V ( T ) | +1 ,s +2) .Suppose to the contrary that either V ( M ) ∩ X V ( T j,t ) Y ( i, | V ( T ) | +1 ,s +2) , or A L ( i, | V ( T ) | +1 ,s +2) ( V ( M ) ∩ X V ( T j,t ) ) ∩ X V ( T j,t ) = ∅ . Note that V ( M ) ∩ X V ( T j,t ) ⊆ W ( i )4 . ByClaim 4.12.1, some component Q of G [ V ( M ) ∩ X V ( T j,t ) ] is disjoint from W ( i, | V ( T ) | )3 . Since V ( M ) intersects X t ∗ ⊆ X ∂T j,t ∪ X t , V ( Q ) intersects X ∂T j,t ∪ X t . Since X t ∩ I j ⊆ W ( i, − , V ( Q ) ∩ X t = ∅ . So V ( Q ) intersects X ∂T j,t . Since X t ∗ ∩ I j ⊆ W ( i, | V ( T ) | )3 , there exist a node t ′ ∈ ∂T j,t with X t ′ ∩ V ( Q ) = ∅ and X t ′ ∩ I j W ( i, | V ( T ) | )3 and a path P in M internally disjointfrom W ( i, | V ( T ) | )3 ∪ V ( Q ) passing through a vertex in V ( M ) ∩ W ( i, | V ( T ) | )3 , a vertex in X t ′ − V ( Q ) and a vertex in X t ′ ∩ V ( Q ) in the order listed. Furthermore, choose Q such that P is as shortas possible.Let P , P , . . . , P m (some some m ∈ N ) be the maxi-mal subpaths of P contained in G [ X V ( T j,t ) ] internally disjoint from64 ∂T j,t ∪ X t , where P intersects W ( i, | V ( T ) | )3 and P passes through P , P , . . . , P m in the order listed. So P m intersects Q . Since P is internally disjointfrom W ( i, | V ( T ) | )3 ⊇ X t , there exist (not necessarily distinct) t , t , . . . , t m +1 ∈ ∂T j,t such thatfor every ℓ ∈ [ m ] , P ℓ is from X t ℓ to X t ℓ +1 . So V ( Q ) ∩ X t m +1 = ∅ .Since V ( Q ) ∩ W ( i, | V ( T ) | )3 = ∅ , there exists a minimum ℓ ∗ ∈ [ m + 1] such that X t ℓ ∗ ∩ I j W ( i, | V ( T ) | )3 . So X t ℓ ∗− ∩ I j ⊆ W ( i, | V ( T ) | )3 . Let t = t ′ . Note that if there exists q ∈ [ − , | V ( T ) | ] such that I j ∩ W ( i,q )3 = I j ∩ W ( i,q +1)3 , then I j ∩ W ( i,q )3 = I j ∩ W ( i, | V ( T ) | )3 . This implies that if q ∈ [ − , | V ( T ) | ] is the minimum such that I j ∩ W ( i,q )3 = I j ∩ W ( i, | V ( T ) | )3 , then there exist atleast q nodes t ′′ ∈ ∂T j,t such that X t ′′ ∩ I j ⊆ W ( i,q )3 . This together with X t ℓ ∗ ∩ I j W ( i, | V ( T ) | )3 imply that X t ℓ ∗− ∩ I j W ( i, | V ( T ) |− .Since P ℓ ∗ − is contained in G [ W ( i )4 ] and intersects X t ℓ ∗− , and X t ℓ ∗− ∩ I j ⊆ W ( i, | V ( T ) |− ,Claim 4.12.1 implies that V ( P ℓ ∗ − ) ⊆ Y ( i, | V ( T ) | ,k ) , where k + 1 is the color of M . Let v bethe vertex in V ( P ℓ ∗ − ) ∩ X t ℓ ∗ . Since X t ℓ ∗ ∩ I j S | V ( T ) | ℓ = − W ( i,ℓ )3 , v ∈ ( D ( i, | V ( T ) | , − X t ) ∪ { u ∈ D ( i, | V ( T ) | , ∩ X t ∩ X q ′ ∩ I j : q ′ ∈ V ( T t ) − { t } , q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 for some i ′ ∈ [0 , i − and ℓ ′ ∈ [0 , | V ( T ) | ] } .Suppose there exists q ′ ∈ V ( T t ) − { t } such that v ∈ D ( i, | V ( T ) | , ∩ X t ∩ X q ′ ∩ I j and q ′ is awitness for X q ′ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 for some i ′ ∈ [0 , i − and ℓ ′ ∈ [0 , | V ( T ) | ] . Then there exists anode t ′′ of T with σ T ( t ′′ ) = i ′ ∈ [0 , i − and with t ∈ V ( T t ′′ ) − { t ′′ } such that q ′ ∈ ∂T j,t ′′ is awitness for X q ′ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 for some ℓ ′ ∈ [0 , | V ( T ) | ] . Since q ′ ∈ ( V ( T t ) − { t } ) ∩ ∂T j,t ′′ , weknow ∂T j,t ′′ is disjoint from the path in T between t ′′ and t . So t ℓ ∗ ∈ ∂T j,t ∩ V ( T j,t ′′ ) . Hence t ′′ is a node of T with t ∈ V ( T t ′′ ) − { t ′′ } and V ( T j,t ′′ ) ∩ ∂T j,t = ∅ such that v ∈ V ( M ) ∩ X q ′ = ∅ and q ′ is a witness for X q ′ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 for some ℓ ′ ∈ [ − , | V ( T ) | ] , a contradiction.Therefore no such q ′ exists, and hence v ∈ D ( i, | V ( T ) | , − X t . So v ∈ W ( i ′ ,ℓ ′ +1 ,k )0 for some i ′ ∈ [0 , i ] , ℓ ′ ∈ [0 , | V ( T ) | ] with ( i ′ , ℓ ′ ) lexicographically at most ( i, | V ( T ) | − . We assume that ( i ′ , ℓ ′ ) is lexicographically minimum. So there exists a monochromatic path P ′ in G [ Y ( i ′ ,ℓ ′ +1 ,k ) ∩ W ( i ′ )4 ] from v to W ( i ′ ,ℓ ′ )3 internally disjoint from W ( i ′ ,ℓ ′ )3 . The minimality of ( i ′ , ℓ ′ ) implies that v D ( i ′ ,ℓ ′ , . So if i ′ = i , then t ℓ ∗ is a witness for X t ℓ ∗ ∩ I j ⊆ W ( i,ℓ ′ +1)3 ⊆ W ( i, | V ( T ) | )3 , acontradiction. Hence i ′ < i .Let u be the end of P ′ in W ( i ′ ,ℓ ′ )3 . Since P ′ is in G [ Y ( i ′ ,ℓ ′ +1 ,k ) ∩ W ( i ′ )4 ] from v to W ( i ′ ,ℓ ′ )3 internally disjoint from W ( i ′ ,ℓ ′ )3 , there exists q ∗ ∈ ∂T j,t ′′ ∪ { t ′′ } , where t ′′ is the node with σ T ( t ′′ ) = i ′ , such that u ∈ X q ∗ ∩ I j and q ∗ is a witness for X q ∗ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 . Since i ′ < i and v ∈ ( X V ( T t ) − X t ) ∩ W ( i ′ )4 , we know that t ∈ V ( T t ′′ ) − { t ′′ } and ∂T j,t ′′ is disjoint from the pathin T between t ′′ and t . So V ( T j,t ′′ ) ∩ ∂T j,t = ∅ . But u ∈ V ( M ) ∩ X q ∗ and q ∗ ∈ ∂T j,t ′′ ∪ { t ′′ } isa witness for X q ∗ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 with i ′ < i , a contradiction. This proves the claim.For every node t of T , let i t be σ T ( t ) .Given any c -monochromatic component M with S M ∩ I ◦ j = ∅ for some j ∈ [ |V| − , definethe following. • Define K ( M ) to be the subset of V ( T ) constructed by repeatedly applying the followingprocess until no more nodes can be added: – r M ∈ K ( M ) . 65 For every t ∈ K ( M ) , if there exists t ′ ∈ ∂T j,t such that V ( M ) ∩ X V ( T t ′ ) − X t ′ = ∅ ,then adding t ′ into K ( M ) . • Define K ∗ ( M ) := { r M }∪{ t ′ ∈ K ( M ) −{ r M } : t is the node in K ( M ) such that t ′ ∈ ∂T j,t ,and | V ( M ) ∩ Y ( i t ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ′ ) | > | V ( M ) ∩ Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T t ′ ) |} . Claim 4.12.26. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∩ I ◦ j = ∅ .Then | V ( M ) | | K ∗ ( M ) | · η .Proof. For every t ∈ K ( M ) , since t is a witness for X t ∩ I j ⊆ W ( i t , − , by Claim 4.12.25, | V ( M ) ∩ X V ( T j,t ) | = | V ( M ) ∩ Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T j,t ) | . So | V ( M ) | = P t ∈ K ( M ) | V ( M ) ∩ X V ( T j,t ) | = P t ∈ K ( M ) | V ( M ) ∩ Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T j,t ) | . Note that for every t ∈ K ( M ) −{ r M } ,there exists k t ∈ K ( M ) with t ∈ V ( T k t ) such that | V ( M ) ∩ Y ( i kt , | V ( T ) | +1 ,s +2) ∩ X V ( T j,t ) | = | V ( M ) ∩ Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T j,t ) | (since t is a candidate for k t ). Choose such a node k t with height as small as possible. Thus k t ∈ K ∗ ( M ) for each t ∈ K ( M ) − { r M } . Let K ′ = { r M } ∪ { t ′ ∈ V ( T ) : t ′ = k t for some t ∈ K ( M ) − { r M }} . Note that K ′ ⊆ K ∗ ( M ) . So | V ( M ) | = X t ∈ K ( M ) | V ( M ) ∩ Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T j,t ) | = | V ( M ) ∩ Y ( i rM , | V ( T ) | +1 ,s +2) ∩ X V ( T j,rM ) | + X t ∈ K ( M ) −{ r M } | V ( M ) ∩ Y ( i kt , | V ( T ) | +1 ,s +2) ∩ X V ( T j,t ) | X q ∈ K ′ | V ( M ) ∩ Y ( i q , | V ( T ) | +1 ,s +2) ∩ X V ( T q ) | X q ∈ K ∗ ( M ) | V ( M ) ∩ Y ( i q , | V ( T ) | +1 ,s +2) ∩ X V ( T q ) | | K ∗ ( M ) | · η by Claim 4.12.24. Claim 4.12.27. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∩ I ◦ j = ∅ .Let t ∈ V ( T ) with V ( M ) ∩ X t = ∅ . Then either V ( M ) ∩ X V ( T t ) ⊆ Y ( i t , − , , or thereexists a monochromatic component M ′ in G [ Y ( i t , − , ] with M ′ ⊆ M , V ( M ′ ) ∩ X t = ∅ and A L ( it, − , ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ .Proof. We may assume V ( M ) ∩ X V ( T t ) Y ( i t , − , , for otherwise we are done. Since X t ⊆ Y ( i t , − , , there exists a path P in M from X t to a vertex v ∈ V ( M ) ∩ X V ( T t ) − Y ( i t , − , internally disjoint from X t such that V ( P − v ) ⊆ Y ( i t , − , . Hence the monochromatic com-ponent M ′ in G [ Y ( i t , − , ] containing P − v satisfies that M ′ ⊆ M , V ( M ′ ) ∩ X t = ∅ and v ∈ A L ( it, − , ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ . Claim 4.12.28. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∩ I ◦ j = ∅ .Let t ∈ K ( M ) . If K ∗ ( M ) ∩ V ( T t ) − { t } 6 = ∅ , then there exists a monochromatic component M ′ in G [ Y ( i t , − , ] with M ′ ⊆ M , V ( M ′ ) ∩ X t = ∅ and A L ( it, − , ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ .Proof. Suppose to the contrary. Since t ∈ K ( M ) , V ( M ) ∩ X t = ∅ . So by Claim 4.12.27, V ( M ) ∩ X V ( T t ) ⊆ Y ( i t , − , . But K ∗ ( M ) ∩ V ( T t ) − { t } 6 = ∅ , there exist t , t with t ∈ ∗ ( M ) ∩ V ( T t ) − { t } and t ∈ K ( M ) with t ∈ ∂T j,t . Since t ∈ K ( M ) and t ∈ K ∗ ( M ) ∩ V ( T t ) − { t } , t ∈ V ( T t ) . Since t ∈ K ∗ ( M ) − { r M } , V ( M ) ∩ Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) = V ( M ) ∩ Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T t ) . But i t > i t > i t and X V ( T t ) ⊆ X V ( T t ) , a contradiction.For any i ∈ [0 , | V ( T ) | − , ℓ ∈ [ − , | V ( T ) | +1] , k ∈ [0 , s +2] and j ∈ [ |V| − , we define the ( i, ℓ, k, j ) -signature to be the sequence ( a , a , . . . , a | V ( G ) | ) such that for every α ∈ [ | V ( G ) | ] , • if σ ( M ) = α for some monochromatic component M in G [ Y ( i,ℓ,k ) ] intersecting X t with A L ( i,ℓ,k ) ( V ( M )) ∩ X V ( T t ) − X t = ∅ (where t is the node of T with i t = i ) andcontained in some s -segment in S ◦ j whose level equals the color of M , then define a α = | A L ( i,ℓ,k ) ( V ( M )) ∩ X V ( T t ) − X t | , • otherwise, define a α = 0 .In the case a α > , we say that M defines a α . Claim 4.12.29. Let i ∈ [0 , | V ( T ) | − , ℓ ∈ [ − , | V ( T ) | + 1] , k ∈ [0 , s + 2] and j ∈ [ |V| − .Then the ( i, ℓ, k, j ) -signature has at most w nonzero entries, and every entry is at most f ( η ) .Proof. Let t be the node of T with i t = i . Since S S ∈S ◦ j S ⊆ I j and | X t ∩ I j | w , there are atmost w monochromatic components M in G [ Y ( i,ℓ,k ) ] intersecting X t with A L ( i,ℓ,k ) ( V ( M )) ∩ X V ( T t ) − X t = ∅ and contained in some s -segment in S ◦ j whose level equals the color of M .So the ( i, ℓ, k, j ) -signature has at most w nonzero entries.Let a α be a nonzero entry of the ( i, ℓ, k, j ) -signature. Let M be the monochromaticcomponent defining a α . So a α = | A L ( i,ℓ,k ) ( V ( M )) ∩ X V ( T t ) − X t | | N > sG ( Y ( i,ℓ,k ) ∩ I j ∩ X V ( T t ) ) ∩ X V ( T t ) | | N > sG ( Y ( i, | V ( T ) | +1 ,s +2) ∩ I j ∩ X V ( T t ) ) ∩ X V ( T t ) | f ( η ) by Claim 4.12.24.For any i ∈ [0 , | V ( T ) | − , ℓ ∈ [ − , | V ( T ) | + 1] , k ∈ [0 , s + 2] and j ∈ [ |V| − , wedefine the ( i, ℓ, k, j ) -pseudosignature to be the sequence ( p , p , . . . , p | V ( G ) | ) such that for every α ∈ [ | V ( G ) | ] , p α is a sequence of length | V ( G ) | such that • if σ ( M ) = α for some monochromatic E ( i ) j,t -pseudocomponent M in G [ Y ( i,ℓ,k ) ] intersect-ing X t with A L ( i,ℓ,k ) ( V ( M )) ∩ X V ( T t ) − X t = ∅ (where t is the node of T with i t = i ) andcontained in some s -segment in S ◦ j whose level equals the color of M , then for every β ∈ [ | V ( G ) | ] , – the β -th entry of p α is the β -th entry of the ( i, ℓ, k, j ) -signature if the β -th entryof the ( i, ℓ, k, j ) -signature is positive and the monochromatic component definingthe β -th entry of the ( i, ℓ, k, j ) -signature is contained in M , – otherwise, the β -th entry of p α is 0, • otherwise, p α is a zero sequence.In the case p α is not a zero sequence, we say that M defines p α . Claim 4.12.30. Let i ∈ [0 , | V ( T ) |− , ℓ ∈ [ − , | V ( T ) | +1] , k ∈ [0 , s +2] and j ∈ [ |V|− . Let ( p , p , . . . , p | V ( G ) | ) be the ( i, ℓ, k, j ) -pseudosignature. Then for every α ∈ [ | V ( G ) | ] , there existat most one index β such that the α -th entry of p β is nonzero. Furthermore, the ( i, ℓ, k, j ) -pseudosignature has at most w nonzero entries, and every entry is at most f ( η ) . roof. Every monochromatic component in G [ Y ( i,ℓ,k ) ] is contained in at most one E ( i ) j,t -pseudocomponent in G [ Y ( i,ℓ,k ) ] . So for every α ∈ [ | V ( G ) | ] , there exist at most one index β such that the α -th entry of p β is nonzero. Hence the number of nonzero entries and thesum of the entries of the ( i, ℓ, k, j ) -pseudosignature equal the number of nonzero entries andthe sum of the entries of the ( i, ℓ, k, j ) -signature, respectively. Then this claim follows fromClaim 4.12.29.For i ∈ N , j ∈ [ |V| − , ℓ ∈ [ − , | V ( T ) | + 1] and k ∈ [0 , s + 2] , we say a monochromaticsubgraph M in G [ Y ( i,ℓ,k ) ] is S ◦ j -related if V ( M ) is contained in some s -segment in S ◦ j whoselevel equals the color of M . Claim 4.12.31. Let i , i ∈ N . Let j ∈ [ |V| − . Let ℓ , ℓ ∈ [ − , | V ( T ) | + 1] . Let k , k ∈ [0 , s + 2] . Let t , t be nodes of T with σ T ( t ) = i and σ T ( t ) = i . Assume that t ∈ V ( T t ) and ( i , ℓ , k ) is lexicographically smaller than ( i , ℓ , k ) . Let M be an S ◦ j -relatedmonochromatic E ( i ) j,t -pseudocomponent in G [ Y ( i ,ℓ ,k ) ] intersecting X t . Then there exist amonochromatic E ( i ) j,t -pseudocomponent M in G [ Y ( i ,ℓ ,k ) ] such that M ⊆ M , σ ( M ) = σ ( M ) and V ( M ) ∩ X t = ∅ . Furthermore, if A L ( i ,ℓ ,k ( V ( M )) ∩ X V ( T t ) − X t = ∅ , then A L ( i ,ℓ ,k ( V ( M )) ∩ X V ( T t ) − X t = ∅ .Proof. Let v M be the vertex of M such that σ ( v M ) = σ ( M ) . Since V ( M ) ∩ X t = ∅ , t ∈ V ( T r vM ) . So there exists a monochromatic component M ′ in G [ Y ( i ,ℓ ,k ) ] containing v M .Hence there exists a monochromatic E ( i ) j,t -pseudocomponent M in G [ Y ( i ,ℓ ,k ) ] containing M ′ . So σ ( M ) σ ( M ′ ) σ ( v M ) = σ ( M ) . Since t ∈ V ( T t ) , E ( i ) j,t ⊆ E ( i ) j,t ⊆ E ( i ) j,t . Since ( i , ℓ , k ) is lexicographically smaller than ( i , ℓ , k ) , M ⊆ M . Hence σ ( M ) = σ ( M ) .Suppose V ( M ) ∩ X t = ∅ . Since t ∈ V ( T r vM ) and v M ∈ V ( M ) , V ( M ) ∩ X V ( T t ) = ∅ .Since V ( M ) ∩ X t = ∅ , there exists a path P in M from X t to V ( M ) internally disjointfrom V ( M ) ∪ X t . Let u be the vertex in V ( P ) ∩ V ( M ) . Since V ( M ) ⊆ Y ( i ,ℓ ,k ) and t ∈ V ( T t ) , the edge of P incident with u belongs to E ( i ) j,t − E ( i ) j,t . But u ∈ V ( M ) , so u X V ( T t ) . Hence there exists no element in E ( i ) j,t − E ( i ) j,t containing u , a contradiction.Therefore V ( M ) ∩ X t = ∅ .Now we assume that A L ( i ,ℓ ,k ( V ( M )) ∩ X V ( T t ) − X t = ∅ . Suppose that A L ( i ,ℓ ,k ( V ( M )) ∩ X V ( T t ) − X t = ∅ . Since ( i , ℓ , k ) is lexicographically smaller than ( i , ℓ , k ) and X V ( T t ) − X t ⊆ X V ( T t ) − X t , V ( M ) ⊃ V ( M ) . Since A L ( i ,ℓ ,k ( V ( M )) ∩ X V ( T t ) − X t = ∅ , there exists e ∈ E ( i ) j,t − E ( i ) j,t such that e ∩ V ( M ) = ∅ 6 = e − V ( M ) . Since t ∈ V ( T t ) , E ( i ) j,t = E ( i ) j,t , so E ( i ) j,t − E ( i ) j,t ⊆ E ( i ) j,t − E ( i ) j,t . So there exists i ∈ [ i , i − and t ∈ V ( T ) with σ T ( t ) = i and t ∈ V ( T t ) such that e ∈ E ( i +1) j,t − E ( i ) j,t . We may assume i is as small as possible. Hence there exist t ∈ ∂T j,t and α ∈ [0 , | V ( G ) | ] such that t is thewitness for e ∈ E ( i ,α +1) j,t − E ( i ,α ) j,t .Since A L ( i ,ℓ ,k ( V ( M )) ∩ X V ( T t ) − X t = ∅ , by the minimality of i , M is a monochromatic E ( i t ,α ) j,t -pseudocomponent in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] and A L ( i , | V ( T ) | +1 ,s +2) ( V ( M )) ∩ X V ( T t ) − X t = ∅ . Since i > i , t ∈ V ( T t ) . So A L ( i ,ℓ ,k ( V ( M )) ∩ X V ( T t ) − X t ⊇ A L ( i , | V ( T ) | +1 ,s +2) ( V ( M )) ∩ X V ( T t ) − X t = ∅ , a contradiction. This proves the claim.68 laim 4.12.32. Let t , t , t be nodes of T such that T t ⊆ T t ⊂ T t . For α ∈ [3] , let ( a ( α )1 , a ( α )2 , . . . , a ( α ) | V ( G ) | ) be the ( i t α , − , , j ) -pseudosignature. Let α ∗ be the largest index suchthat a (1) α ∗ is a nonzero sequence. Then the following hold. • For every β ∈ [ α ∗ − , if a (1) β is a zero sequence, then a (2) β is a zero sequence. • For every β ∈ [ α ∗ ] , if a (2) β is a zero sequence, then a (3) β is a zero sequence.Proof. We first prove the first statement. Suppose there exists β ∈ [ α ∗ − such that a (2) β is a nonzero sequence. Let M be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining a (2) β . Since a (1) α ∗ is a nonzero sequence, there exists a monochromatic E ( i t ) j,t -pseudocomponent M ′ in G [ Y ( i t , − , ] defining a (1) α ∗ . Since σ ( M ) = β < α ∗ = σ ( M ′ ) and V ( M ′ ) ∩ X t = ∅ 6 = V ( M ) ∩ X t , V ( M ) ∩ X t = ∅ . By Claim 4.12.31, there exists amonochromatic E ( i t , − , j,t -pseudocomponent M such that M ⊆ M , σ ( M ) = σ ( M ) = β and A L ( it , − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ . So a (1) β is a nonzero sequence, a contradiction.Now we prove the second statement. Suppose to the contrary that a (2) β is a zero sequenceand a (3) β is a nonzero sequence. If β = α ∗ , then a (1) β is a nonzero sequence; if β ∈ [ α ∗ − ,then a (1) β is a nonzero sequence by the first statement of this claim. Since a (3) β is a nonzerosequence, there exists a monochromatic E ( i t ) j,t -pseudocomponent Q in G [ Y ( i t , − , ] defining a (3) β . Since a (1) β is a nonzero sequence, there exists a monochromatic E ( i t ) j,t -pseudocomponent Q in G [ Y ( i t , − , ] defining a (1) β . Hence σ ( Q ) = β = σ ( Q ) . So Q ⊆ Q . Hence V ( Q ) ∩ X t = ∅ 6 = V ( Q ) ∩ X t . So V ( Q ) ∩ X t = ∅ . By Claim 4.12.31, there exists amonochromatic E ( i t , − , j,t -pseudocomponent Q such that Q ⊆ Q , σ ( Q ) = σ ( Q ) = β and A L ( it , − , ( V ( Q )) ∩ X V ( T t ) − X t = ∅ . So a (2) β is a nonzero sequence, a contradiction. Claim 4.12.33. Let j ∈ [ |V| − . Let t ∈ V ( T ) . Let M be an S ◦ j -related monochro-matic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ ∈ V ( T ) be the node with t ∈ V ( T j,t ∗ ) such that some monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] in-tersecting V ( M ) intersects X t ′ for some witness t ′ ∈ ( ∂T j,t ∗ ) ∪ { t ∗ } for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 forsome ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i t ∗ is minimum. Then for every monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent M in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) , • M intersects X t ′′ for some witness t ′′ ∈ ( ∂T j,t ∗ ) ∪ { t ∗ } for X t ′′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and • for every node q ∈ ( ∂T j,t ∗ ) ∪{ t ∗ } with V ( M ) ∩ X q = ∅ , q is a witness for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] .Proof. Note that t ∗ exists since t is an candidate. So i t ∗ i t .If there exist distinct nodes q , q ∈ ( ∂T j,t ∗ ) ∪ { t ∗ } where q is a witness for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] such that there exists a path P in M [ V ( M ) ∩ X V ( T j,t ∗ ) ] from X q to X q internally disjoint from X ∂T j,t ∗ , then by the minimality of i t ∗ and the constructionof E ( i t ) j,t , E ( P ) ∩ E ( i t ) j,t = ∅ , so P is a path in G , and hence by Claim 4.12.25, V ( P ) ⊆ Y ( i t ∗ , | V ( T ) | +1 ,s +2) . 69ence, since | ∂T j,t ∗ | | V ( T ) | − and some monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) intersects X t ′′ for some witness t ′′ ∈ ( ∂T j,t ∗ ) ∪ { t ∗ } for X t ′′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , to prove this claim, it suffices to prove thatif M is a component of M [ V ( M ) ∩ X V ( T j,t ∗ ) ] such that V ( M ) ∩ X t ′′ = ∅ for some witness t ′′ ∈ ( ∂T j,t ∗ ) ∪ { t ∗ } for X t ′′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , then for every node q ∈ ( ∂T j,t ∗ ) ∪ { t ∗ } with V ( M ) ∩ X q = ∅ , q is a witness for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] .Let M be a component of M [ V ( M ) ∩ X V ( T j,t ∗ ) ] such that V ( M ) ∩ X t ′′ = ∅ for somewitness t ′′ ∈ ( ∂T j,t ∗ ) ∪ { t ∗ } for X t ′′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] . Since M is S ◦ j -related, V ( M ) ⊆ W ( i t ∗ )4 . By the minimality of i t ∗ , E ( M ) ∩ E ( i t ) j,t = ∅ .Suppose to the contrary that there exists q ∈ ( ∂T j,t ∗ ) ∪{ t ∗ } such that V ( M ) ∩ X q = ∅ , and q is not a witness for X q ∩ I j ⊆ W ( i t ∗ ,α )3 for any α ∈ [ − , | V ( T ) | ] . So ℓ | ∂T j,t ∗ |− | V ( T ) |− .Hence ℓ ∈ [ − , | V ( T ) | − . We may choose q and t ′′ such that there exists v ∈ V ( M ) ∩ X q such that there exists a monochromatic path P ∗ in M from X t ′′ to X q internally disjointfrom X t ∗ ∪ X ∂T j,t ∗ . Recall that P ∗ is a monochromatic path in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] by theminimality of i t ∗ and Claim 4.12.25.Let k +1 be the color of M . By Claim 4.12.1, V ( P ∗ ) ⊆ Y ( i t ∗ ,ℓ +1 ,k ) . Since q is not a witnessfor X q ∩ I j ⊆ W ( i t ∗ ,ℓ +1)3 , either v ∈ D ( i t ∗ ,ℓ +1 , − X t ∗ , or v ∈ D ( i t ∗ ,ℓ +1 , ∩ X t ∗ ∩ X q ′ ∩ I j for somewitness q ′ ∈ V ( T t ∗ ) − { t ∗ } for X q ′ ∩ I j ⊆ W ( i ′ ,ℓ ′ )3 for some i ′ ∈ [0 , i t ∗ − and ℓ ′ ∈ [0 , | V ( T ) | ] .The latter cannot hold by the minimality of i t ∗ .Hence v ∈ D ( i t ∗ ,ℓ +1 , − X t ∗ . By the minimality of i t ∗ , there exists a minimum ℓ ∗ ∈ [0 , | V ( T ) | − such that v ∈ D ( i t ∗ ,ℓ ∗ +1 , . So v ∈ W ( i t ∗ ,ℓ ∗ +1 ,k )0 . Hence there exists a path in G [ Y ( i t ∗ ,ℓ ∗ +1 ,k ) ∩ W ( i t ∗ )4 ] with color k +1 from v to W ( i t ∗ ,ℓ ∗ )3 . So q is a witness for X q ∩ I j ⊆ W ( i t ∗ ,ℓ ′ )3 for some ℓ ′ ∈ [ − , ℓ ∗ + 1] , a contradiction. This proves the claim. Claim 4.12.34. Let i ∈ N and t ∈ V ( T ) with σ T ( t ) = i . Let j ∈ [ |V| − . Let ( p , p , . . . , p | V ( G ) | ) be the ( i, − , , j ) -pseudosignature. Let α ∈ [ | V ( G ) | ] be the smallest in-dex such that p α is not a zero sequence. Let M be the monochromatic E ( i ) j,t -pseudocomponentdefining p α . Let i ′ ∈ N and t ′ ∈ V ( T t ) − { t } with σ T ( t ′ ) = i ′ . Let M ′ be the monochromatic E ( i ′ ) j,t ′ -pseudocomponent in G [ Y ( i ′ , − , ] containing M . Then V ( M ′ ) = V ( M ) ⊆ Y ( i, − , .Proof. Since V ( M ) ⊆ Y ( i, − , , it suffices to show that V ( M ′ ) = V ( M ) . Suppose to thecontrary that V ( M ′ ) = V ( M ) . We choose i, i ′ such that i ′ − i is minimum among all coun-terexamples.Since V ( M ′ ) = V ( M ) , there exists the minimum i ′′ ∈ [ i, i ′ ] such that V ( M ′′ ) = V ( M ) ,where M ′′ is the monochromatic E ( i ′′ ) j,t ′′ -pseudocomponent in G [ Y ( i ′′ , − , ] containing M and t ′′ is the node with σ T ( t ′′ ) = i ′′ . Note that i ′′ > i . Let p be the parent of t ′′ . So p ∈ V ( T t ) .For every node z with z ∈ V ( T t ) , let M z be the monochromatic E ( i z ) j,z -pseudocomponentin G [ Y ( i z , − , ] containing M .Suppose there exist z ∈ V ( T ) with T p ⊆ T z ⊆ T t and an S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M ′ z in G [ Y ( i z , − , ] with A L ( iz, − , ( V ( M ′ z )) ∩ X V ( T z ) − X z = ∅ intersecting X z such that σ ( M ′ z ) < σ ( M z ) . By the minimality of i ′′ , V ( M z ) = V ( M ) . Since M z contains70 , V ( M z ) ∩ X t = ∅ . Since σ ( M ′ z ) < σ ( M z ) and V ( M ′ z ) ∩ X z = ∅ , V ( M ′ z ) ∩ X t = ∅ . ByClaim 4.12.31, there exists α ′ ∈ [ α − such that p α ′ is not a zero sequence, a contradiction.So for every z ∈ V ( T ) with T p ⊆ T z ⊆ T t , there exists no S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M ′ z in G [ Y ( i z , − , ] with A L ( iz, − , ( V ( M ′ z )) ∩ X V ( T z ) − X z = ∅ intersecting X z such that σ ( M ′ z ) < σ ( M z ) .Hence V ( Q ) = V ( M p ) = V ( M ) , where Q is the monochromatic E ( i p ) j,p -pseudocomponent in G [ Y ( i p , | V ( T ) | +1 ,s +2) ] containing M p . Since V ( M ′′ ) = V ( M ) and t ′′ ∈ V ( T j,p ) , by the minimalityof i ′′ and Claim 4.12.25, there exists e ∈ E ( M ′′ ) ∩ E ( i ′′ ) j,t ′′ − E ( i p ) j,p such that e ∩ V ( Q ) = ∅ 6 = e − V ( Q ) .Since E ( i p ) j,t ′′ = E ( i p ) j,p . So V ( Q ) intersects an element in E ( i ′′ ) j,t ′′ − E ( i p ) j,t ′′ . Then since t ′′ is achild of p , there exist q ∈ ∂T j,p and an S ◦ j -related monochromatic E ( i p ) j,t ′′ -pseudocomponent Q ′ in G [ Y ( i p , | V ( T ) | +1 ,s +2) ] with A L ( ip, | V ( T ) | +1 ,s +2) ( V ( Q ′ )) ∩ X V ( T p ) − X p = ∅ intersecting X q suchthat σ ( Q ′ ) < σ ( Q ) . Since V ( Q ) = V ( M ) , σ ( Q ′ ) < σ ( Q ) = σ ( M ) . Since V ( Q ) ∩ X t = ∅ and V ( Q ′ ) ∩ X q = ∅ , V ( Q ′ ) ∩ X t = ∅ . Let Q ′′ be the E ( i ′′ ) j,t ′′ -pseudocomponent in G [ Y ( i ′′ , − , ] containing Q ′ . Then σ ( Q ′′ ) σ ( Q ′ ) < σ ( M ) . By Claim 4.12.31, there exists α ′ ∈ [ α − such that p α ′ is not a zero sequence and is defined by a subgraph of Q ′′ , a contradiction. Thisproves the claim.For any j ∈ [ |V| − , t ∈ V ( T ) and S ◦ j -related monochromatic E ( i t ) j,t -pseudocomponent M in G [ Y ( i t , − , ] intersecting X t , we say that ( t, M ) is a nice pair if the following hold: • If t ∗ ∈ V ( T ) be the node of T such that t ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponentin G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i t ∗ is minimum, theneither – t = t ∗ , or – t = t ∗ and for every c -monochromatic path P in G intersecting V ( M ) with V ( P ) ⊆ X V ( T t ∗ ) internally disjoint from X V ( T t ) and for every vertex u P of P in X V ( T j,t ∗ ) , there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] such that V ( M ′ ) ∩ V ( M ) = ∅ 6 = V ( M ′ ) ∩ X V ( T t ) , M ′ contains u P , and if A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ , then A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ . Claim 4.12.35. Let j ∈ [ |V| − . Let t ∈ V ( T ) and let M be an S ◦ j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ ∈ V ( T ) be the node of T suchthat t ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , andsubject to this, i t ∗ is minimum. If t ∗ = t , A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ and ( t, M ) is anice pair, then every monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) intersects X V ( T t ) and satisfies A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ .Proof. Let S = { Q : Q is a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] such that V ( Q ) intersects V ( M ) and X V ( T t ) , and A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( Q )) ∩ V ( T t ) − X t = ∅} . Let S = { Q S : Q is a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponentin G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) } . By Claims 4.12.25 and 4.12.33, S = ∅ .Suppose that the claim does not hold. That is, S = ∅ . By the construction of E ( i t ) j,t andClaims 4.12.25 and 4.12.33, there exist monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponents M and M in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) such that M ∈ S , M ∈ S and there exist q ∈ ∂T j,t ∗ with V ( M ) ∩ X q = ∅ 6 = V ( M ) ∩ X q and a path P in M [ V ( M ) ∩ X V ( T q ) ] from V ( M ) ∩ X q to V ( M ) ∩ X q internally disjoint from X q ∪ V ( M ) ∪ M . By the minimality of i t ∗ , P is a c -monochromatic path in G . So V ( P ) ∩ A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M )) ∩ X V ( T q ) − X q = ∅ 6 = V ( P ) ∩ A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M )) ∩ X V ( T q ) − X q . Since either V ( M ) ∩ X V ( T t ) = ∅ or A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M )) ∩ X V ( T t ) − X t = ∅ , q V ( T t ) and P is internally disjoint from X V ( T t ) .Let u be the vertex in V ( P ) ∩ V ( M ) ∩ X q . Since ( t, M ) is nice, there exists M ∈ S suchthat M contains u . Since u ∈ V ( M ) ∩ V ( M ) , M = M . Hence M = M ∈ S , acontradiction. Claim 4.12.36. Let j ∈ [ |V| − . Let t ∈ V ( T ) and let M be an S ◦ j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ ∈ V ( T ) be the node of T suchthat t ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] ,and subject to this, i t ∗ is minimum. If t = t ∗ , A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ and ( t, M ) is a nice pair, then there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) such that σ ( M ∗ ) = σ ( M ) , V ( M ∗ ) ∩ X t = ∅ andsatisfies A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X V ( T t ) − X t = ∅ .Proof. Let v M be the vertex of M such that σ ( v M ) = σ ( M ) . If v M ∈ V ( T t ∗ ) − { t ∗ } , then byClaims 4.12.25 and 4.12.33, there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] containing v M . If v M V ( T t ∗ ) − { t ∗ } , then by Claim 4.12.31, thereexists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] containing v M . Since t = t ∗ , M ∗ ⊆ M . Hence σ ( M ) = σ ( M ∗ ) and V ( M ∗ ) ∩ V ( M ) = ∅ . So byClaim 4.12.35, V ( M ∗ ) ∩ X V ( T t ) = ∅ and A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X V ( T t ) − X t = ∅ . Since V ( M ) ∩ X t = ∅ , t ∈ V ( T r M ) . Since v M ∈ V ( M ∗ ) , V ( M ∗ ) ∩ X r M = ∅ . Since V ( M ∗ ) ∩ X V ( T t ) = ∅ , V ( M ∗ ) ∩ X t = ∅ .For every z ∈ V ( T ) and every S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M in G [ Y ( i z , − , ] intersecting X z , we say that ( z, M ) is an outer-safe pair if the following hold: • If z ∗ ∈ V ( T ) is the node of T such that z ∈ V ( T j,z ∗ ) and some E ( i z ∗ ) j,z ∗ -pseudocomponentin G [ Y ( i z ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,z ∗ ∪ { z ∗ } for X q ∩ I j ⊆ W ( i z ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i z ∗ is minimum, thenthere exist no t ′ ∈ ∂T j,z ∗ − V ( T z ) , u, v ∈ X t ′ , a monochromatic path in G [ Y ( i z , − , ∩ X V ( T t ′ ) ] but not in G [ Y ( i z ∗ , | V ( T ) | +1 ,s +2) ] from u to v internally disjoint from X t ′ such thatthere exists a monochromatic E ( i z ∗ , | V ( G ) | ) j,z ∗ -pseudocomponent M ∗ in G [ Y ( i z ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) = σ ( M ) containing u but not v .(Note that we do not require A L ( iz, − , ( V ( M )) ∩ X V ( T z ) − X z = ∅ .)72 laim 4.12.37. Let j ∈ [ |V| − . Let t ∈ V ( T ) and let M be an S ◦ j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ ∈ V ( T ) be the node of T suchthat t ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] ,and subject to this, i t ∗ is minimum. Assume that t = t ∗ . Assume that ( z, M ′ ) is a nicepair and an outer-safe pair for every z ∈ V ( T ) and every S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M ′ in G [ Y ( i z , − , ] intersecting X z with σ ( M ′ ) < σ ( M ) .Then there exist no t ′ ∈ ∂T j,t ∗ ∩ V ( T t ) , u, v ∈ X t ′ and a monochromatic path in G [ Y ( i t , − , ∩ X V ( T t ′ ) ] but not in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] from u to v internally disjoint from X t ′ such that t ′ isa witness for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) = σ ( M ) containing u butnot v .Proof. Suppose to the contrary that there exist t ′ ∈ ∂T j,t ∗ ∩ V ( T t ) , u ′ , v ′ ∈ X t ′ , a monochro-matic path P ′ in G [ Y ( i t , − , ∩ X V ( T t ′ ) ] internally disjoint from X t ′ , and a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) = σ ( M ) such that t ′ isa witness for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , V ( P ′ ) Y ( i t ∗ , | V ( T ) | +1 ,s +2) , and M ∗ contains u ′ but not v ′ . We further assume that σ ( M ) is minimum among all counterexamples.Let t be the node of T such that i t i t ′ and V ( P ′ ) Y ( i t , − , , and subject to these, i t is maximum. By the maximality of i t , t ′ ∈ V ( T t ) . Since i t ′ > i t > i t ∗ and V ( P ′ ) Y ( i t ∗ , | V ( T ) | +1 ,s +2) , we know V ( P ′ ) Y ( i t ∗ +1 , − , , so i t ∗ < i t . Note that t ′ ∈ ∂T j,t ∗ , so t ′ V ( T j,t ) − ∂T j,t . Let P be the maximal subpath of P ′ [ V ( P ′ ) ∩ Y ( i t , − , ] containing u ′ . Let R be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] containing P . So M ∗ ⊆ R .Let R ′ be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] containing v ′ .Since V ( P ′ ) ⊆ Y ( i t , − , and i t i t ′ , we know i t < i t and t = t ′ . Since t = t ′ and t ′ V ( T j,t ) − ∂T j,t , by the maximality of i t , there exists an S ◦ j -related monochromatic E ( i t ) j,t -pseudocomponent M in G [ Y ( i t , − , ] intersecting X t such that there exist ℓ ′ ∈ [ − , and k ′ ∈ [0 , s + 2] such that σ ( M ) = σ ( Z ) < min { σ ( R ) , σ ( R ′ ) } and A L ( it ,ℓ ′ ,k ′ ) ( V ( Z )) ∩ V ( P ′ ) ∩ N G ( V ( P )) ∩ Z t − Y ( i t , − , = ∅ , where Z is the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t ,ℓ ′ ,k ′ ) ] containing M . So V ( Z ) ∩ X t ′ = ∅ .Since M ∗ ⊆ R , σ ( R ) σ ( M ∗ ) . Hence σ ( M ) < σ ( M ∗ ) = σ ( M ) . Let z ∗ be thenode of T such that t ∈ V ( T j,z ∗ ) and some E ( i z ∗ ) j,z ∗ -pseudocomponent in G [ Y ( i z ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,z ∗ ∪ { z ∗ } for X q ∩ I j ⊆ W ( i z ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i z ∗ is minimum. Since t ∗ is a candidate for z ∗ , i z ∗ i t ∗ .Since σ ( M ) < σ ( M ) , ( t , M ) is a nice pair by assumption. By Claim 4.12.36, there exists amonochromatic E ( i z ∗ , | V ( G ) | ) j,z ∗ -pseudocomponent M ′ in G [ Y ( i z ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) such that σ ( M ′ ) = σ ( M ) , V ( M ′ ) ∩ X t = ∅ and A L ( iz ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ .Since V ( M ) ∩ X t ′ = ∅ and A L ( it , − , ( V ( M )) ∩ X V ( T t ′ ) − X t ′ = ∅ , byClaims 4.12.25 and 4.12.33, there exists a monochromatic E ( i z ∗ , | V ( G ) | ) j,z ∗ -pseudocomponent M in G [ Y ( i z ∗ , | V ( T ) | +1 ,s +2) ] intersecting M such that V ( M ) ∩ X t ′ = ∅ and A L ( iz ∗ , | V ( T ) | +1 ,s +2) ( V ( M )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Suppose M M ′ . Since M ∪ M ′ ⊆ M , there exists a path Q in M V ( M ′ ) to V ( M ) internally disjoint from V ( M ′ ) . By the construction of E ( i t ) j,t andClaims 4.12.25 and 4.12.33, Q is in G [ Y ( i t , − , ∩ X V ( T z ∗ ) ] . Hence there exist t ′′ ∈ ∂T j,z ∗ , u ′′ , v ′′ ∈ X t ′′ , a monochromatic path Q ′ in G [ Y ( i t , − , ∩ X V ( T t ′′ ) ] from u ′′ to v ′′ internallydisjoint from X t ′′ but not in G [ Y ( i z ∗ , | V ( T ) | +1 ,s +2) ] such that M ′ contains u ′′ but not v ′′ . Since σ ( M ) < σ ( M ) , ( t , M ) is an outer-safe pair, so t ′′ ∈ V ( T t ) . But σ ( M ) is minimum amongall counterexamples, so t ′′ V ( T t ) , a contradiction.So M ′ contains M ′ for every monochromatic E ( i z ∗ , | V ( G ) | ) j,z ∗ -pseudocomponent M ′ in G [ Y ( i z ∗ , | V ( T ) | +1 ,s +2) ] intersecting M with that V ( M ′ ) ∩ X t ′ = ∅ and A L ( iz ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Hence there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] containing M ′ such that V ( M ∗ ) ∩ X t ′ = ∅ and A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X V ( T t ′ ) − X t ′ = ∅ .Since M ∗ is an E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent, there exists a nonnegative integer β suchthat M ∗ is the ( β + 1) -smallest among all monochromatic E ( i t ∗ ,β ) j,t ∗ -pseudocomponents in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] . Let R u and R v be the monochromatic E ( i t ∗ ,β ) j,t ∗ -pseudocomponents in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] containing u ′ and v ′ , respectively. By the existence of P ′ , R u ∪ R v ⊆ M ,so σ ( R u ) = σ ( M ∗ ) = σ ( M ) σ ( R v ) .Note that σ ( M ∗ ) σ ( M ′ ) = σ ( M ) < σ ( M ) σ ( R u ) . Since { u, v } 6∈ E ( i t ∗ , | V ( G ) | ) j,t ∗ , { u, v } 6∈ E ( i t ∗ ,β +1) j,t ∗ . Since σ ( M ∗ ) < σ ( R u ) , we know ( V ( R u ) ∪ V ( R v )) ∩ X t ∗ = ∅ and forevery S ◦ j -related monochromatic E ( i t ∗ ,β ) j,t ∗ -pseudocomponent M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ′ ) σ ( M ∗ ) , A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ∗ ) − X t ∗ ⊆ X V ( T t ′ ) − ( X t ′ ∪ Z t ∗ ) .Since ( V ( R u ) ∪ V ( R v )) ∩ X t ∗ = ∅ , we know V ( M ′ ) ∩ X t ∗ = ∅ for every S ◦ j -relatedmonochromatic E ( i t ∗ ,β ) j,t ∗ -pseudocomponent M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ′ ) σ ( M ∗ ) .Since A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ∗ ) − X t ∗ ⊆ X V ( T t ′ ) − ( X t ′ ∪ Z t ∗ ) for every S ◦ j -relatedmonochromatic E ( i t ∗ ,β ) j,t ∗ -pseudocomponent M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ′ ) σ ( M ′ ) ,we know V ( M ) = V ( M ∗ ) . Since σ ( M ∗ ) = σ ( M ) = σ ( Z ) and Z t ∩ X V ( T t ′ ) ⊆ Z t ∗ , we know V ( Z ) = V ( M ) = V ( M ∗ ) and ∅ 6 = A L ( it ,ℓ ′ ,k ′ ) ( V ( Z )) ∩ V ( P ′ ) ∩ N G ( V ( P )) ∩ Z t − Y ( i t , − , ⊆ A L ( it ,ℓ ′ ,k ′ ) ( V ( M ∗ )) ∩ Z t ∩ X V ( T t ′ ) − X t ′ ⊆ A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ Z t ∗ ∩ X V ( T t ′ ) − X t ′ ⊆ A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ ( X V ( T t ∗ ) − X t ∗ ) ∩ Z t ∗ ∩ X V ( T t ′ ) − X t ′ ⊆ ( X V ( T t ′ ) − ( X t ′ ∪ Z t ∗ )) ∩ Z t ∗ ∩ X V ( T t ′ ) − X t ′ = ∅ , a contradiction. Claim 4.12.38. Let j ∈ [ |V|− . Let t ∈ V ( T ) . Let M be an S j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ be the node of T with t ∈ V ( T j,t ∗ ) such that some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) intersects X t ′ for some witness t ′ ∈ ∂T j,t ∗ ∪ { t ∗ } for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , andsubject to this, i t ∗ is minimum. Assume that ( z, M z ) is a nice pair and an outer-safe pairfor every z ∈ V ( T ) and S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M z in G [ Y ( i z , − , ] intersecting X z with σ ( M z ) < σ ( M ) . If ( t, M ) is an outer-safe pair, then the following hold. • There uniquely exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) . • V ( M ∗ ) ∩ X t = ∅ and σ ( M ∗ ) = σ ( M ) . If t ∗ = t and A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ , then A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X V ( T t ) − X t = ∅ .Proof. By Claims 4.12.25, 4.12.31 and 4.12.33, there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) = σ ( M ) such that V ( M ∗ ) intersects X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] .Suppose there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) such that M ′ = M ∗ . Since M ′ ∪ M ∗ ⊆ M ,there exists a monochromatic path P in G [ Y ( i t , − , ] from V ( M ∗ ) to V ( M ′ ) internally disjointfrom V ( M ∗ ) . By tracing P from V ( M ∗ ) , there exist t ′ ∈ ∂T j,t ∗ , u, v ∈ X t ′ , a monochromaticpath Q in G [ Y ( i t , − , ∩ X V ( T t ′ ) ] from u to v such that M ∗ contains u but not v , and V ( Q ) ∩ A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X T t ′ − X t ′ = ∅ . Note that V ( Q ) Y ( i t ∗ , | V ( T ) | +1 ,s +2) , since M ∗ contains u but not v . By Claim 4.12.33, since V ( M ∗ ) ∩ X t ′ = ∅ 6 = V ( M ∗ ) ∩ V ( M ) , t ′ isa witness for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] . Since ( t, M ) is an outer-safe pair, t ′ ∈ V ( T t ) . By Claim 4.12.37, t ′ V ( T t ) , a contradiction.Hence M ∗ is the unique monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) . Recall that σ ( M ∗ ) = σ ( M ) . Since V ( M ) ∩ X t = ∅ and t ∈ V ( T j,t ∗ ) , by Claims 4.12.25 and 4.12.33, V ( M ∗ ) ∩ X t = ∅ .If t ∗ = t and A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ , then since V ( M ) ∩ X t ⊆ V ( M ∗ ) by Claim 4.12.25, and M ∗ is the unique monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) , A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X V ( T t ) − X t = ∅ . Claim 4.12.39. Let j ∈ [ |V| − . Let t ∈ V ( T ) and let M be an S ◦ j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ ∈ V ( T ) be the node of T suchthat t ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] ,and subject to this, i t ∗ is minimum. Assume that t = t ∗ . Assume that ( z, M ′ ) is a nicepair and an outer-safe pair for every z ∈ V ( T ) and every S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M ′ in G [ Y ( i z , − , ] intersecting X z with σ ( M ′ ) < σ ( M ) .If ( t, M ) is an outer-safe pair and there exists a c -monochromatic path P in G [ X V ( T t ) ] from V ( M ) to a monochromatic E ( i t ) j,t -pseudocomponent M in G [ Y ( i t , − , ] other than M internally disjoint from every monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] such that V ( P ) ∩ A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ , then there exist t ′ ∈ ∂T j,t ∗ ∩ V ( T t ) , u, v ∈ X t ′ and a c -monochromatic path in G [ X V ( T t ′ ) ] but not in G [ Y ( i t , − , ] from u to v internally disjoint from X t ′ such that t ′ is a witness for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and there exists amonochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) = σ ( M ) containing u but not v .Proof. Since P is internally disjoint from every monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] and V ( P ) ∩ A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ , V ( P ) ⊆ X V ( T t ) .Since M = M , there exist t ′ ∈ ∂T j,t ∗ ∩ V ( T t ) , u, v ∈ X t ′ , a subpath P ′ of P ∪ M ∪ M from u to v contained in G [ X V ( T t ′ ) ] internally disjoint from X t ′ but not contained in G [ Y ( i t , − , ] ,and a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting75 ( M ) ∩ V ( P ) containing u but not containing v . Note that t ′ is a witness for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] by Claims 4.12.25 and 4.12.33.By Claims 4.12.25, 4.12.31 and 4.12.33, there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent Q ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( Q ∗ ) = σ ( M ) . Suppose σ ( M ∗ ) = σ ( M ) .So M ∗ = Q ∗ . Since M ∗ ∪ Q ∗ ⊆ M , by the construction of E ( i t ) j,t and Claims 4.12.25 and 4.12.33,there exist t ′′ ∈ ∂T j,t ∗ , a monochromatic path Q ′ in G [ Y ( i t , − , ∩ X V ( T t ′′ ) ] from V ( Q ∗ ) to amonochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) otherthan Q ∗ internally disjoint from X t ′′ such that V ( Q ′ ) ∩ A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( Q ∗ )) ∩ X V ( T t ′′ ) − X t ′′ = ∅ , and t ′′ is a witness for X t ′′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] . By Claim 4.12.37, t ′′ V ( T t ) . So ( t, M ) is not an outer-safe pair, a contradiction. Hence σ ( M ∗ ) = σ ( M ) . Thisproves the claim.For any t ∈ V ( T ) , t ′ ∈ V ( T t ) , j ∈ [ |V| − , α ∈ [0 , w ] , α , α ∈ [ w ] with α < α α , k ∈ N , k ′ ∈ N , ℓ ∈ [ − , | V ( T ) | + 1] , ξ ∈ [0 , s + 2] and ξ ′ ∈ [0 , | V ( G ) | ] , we define an ( α , α , α , k, k ′ ) -blocker for ( j, t, ℓ, ξ, ξ ′ , t ′ ) to be a parade ( t − k ′ , t − k ′ +1 , . . . , t − , t , t , . . . , t k ) such that the following hold: • For every β ∈ [ w ] , let M β be the S ◦ j -related monochromatic E ( i t ,ξ ′ ) j,t -pseudocomponentin G [ Y ( i,ℓ,ξ ) ] such that σ ( M β ) is the β -th smallest among all S ◦ j -related monochro-matic E ( i t ,ξ ′ ) j,t -pseudocomponents M in G [ Y ( i t ,ℓ,ξ ) ] intersecting X t ′ with A L ( it,ℓ,ξ ) ( V ( M )) ∩ X V ( T t ′ ) − X t ′ = ∅ (if there are less than β such M , then let M β = ∅ ). • If k ′ = 0 , then t − k ′ ∈ V ( T t ′ ) − { t ′ } . • t ∈ V ( T t ′ ) − { t ′ } . • For every distinct ℓ , ℓ ∈ [ − k ′ , k ] − { } , X t ℓ ∩ I j − X t ′ and X t ℓ ∩ I j − X t ′ are disjointnon-empty sets. • S α β =1 A L ( it,ℓ,ξ ) ( V ( M β )) ∩ X V ( T t ) − X t ⊆ ( X V ( T tk ) − X t k ) ∩ I ◦ j . • There exists α ∗ ∈ { α , α } such that A L ( it,ℓ,ξ ) ( V ( M α ∗ )) ∩ ( X V ( T t ) − X t ) ∩ X V ( T t ) − X t = ∅ ,and if k ′ = 0 , then A L ( it,ℓ,ξ ) ( V ( M α ∗ )) ∩ ( X V ( T t ) − X t ) ∩ X V ( T t − k ′ ) − ( X t − k ′ ∪ I ◦ j ) = ∅ . Claim 4.12.40. Let j ∈ [ |V| − . Let t ∈ V ( T ) and let t ′ ∈ ∂T j,t such that t ′ is a witness for X t ′ ∩ I j ⊆ W ( i t ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] . Let M ′ , M ′ , . . . , M ′ k ′ (for some k ′ ∈ N ) be the S ◦ j -related monochromatic E ( i t ′ ) j,t ′ -pseudocomponents in G [ Y ( i t ′ , − , ] such that for every α ∈ [ k ′ ] , V ( M ′ α ) ∩ X t ′ = ∅ and A L ( it ′ , − , ( V ( M ′ α )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Assume that σ ( M ′ ) < σ ( M ′ ) < · · · < σ ( M ′ k ′ ) . Let V ( M ′ γ ) = ∅ for every γ > k ′ + 1 . Let α ′ ∈ [ w ] and α ′ ∈ [ α ′ + 1 , k ′ ] .Let M (0)1 , M (0)2 , . . . , M (0) k (0) (for some k (0) ∈ N ) be the S ◦ j -related monochro-matic E ( i t ) j,t -pseudocomponents in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] such that for every α ∈ [ k (0) ] , A L ( it, | V ( T ) | +1 ,s +2) ( V ( M (0) α )) ∩ X V ( T t ) − X t = ∅ . Assume that σ ( M (0)1 ) < σ ( M (0)2 ) < · · · <σ ( M (0) k (0) ) . Let β be the minimum such that M (0) β ⊆ M ′ α ′ . Let β ∈ [ | V ( G ) | ] be the inte-ger such that the monochromatic E ( i t ,β − j,t -pseudocomponent in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] containing M (0) β has the β -th smallest σ -value among all monochromatic E ( i t ,β − j,t -pseudocomponentsin G [ Y ( i t , | V ( T ) | +1 ,s +2) ] . Let β be the largest element in [ β − such that the monochro-matic E ( i t ,β − j,t -pseudocomponent M in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] with the β -th smallest σ -value mong all monochromatic E ( i t ,β − j,t -pseudocomponents in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] has the proper-ties that M is S ◦ j -related, V ( M ) ∩ X t ′ = ∅ and A L ( it, | V ( T ) | +1 ,s +2) ( V ( M )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Let M , M , . . . , M k (for some k ∈ N ) be the S ◦ j -related monochromatic E ( i t ,β − j,t -pseudocomponents in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] such that for every α ∈ [ k ] , V ( M α ) ∩ X t ′ = ∅ and A L ( it, | V ( T ) | +1 ,s +2) ( V ( M α )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Assume that σ ( M ) < σ ( M ) < · · · < σ ( M k ) .Let V ( M γ ) = ∅ for every γ > k + 1 .For γ ∈ { , } , let Q γ = { M ℓ : M ℓ ⊆ M ′ α ′ γ } . For every Q ∈ Q ∪ Q , let α Q be the indexsuch that Q = M α Q .Assume V ( M ′ α ′ ) ∩ X t = ∅ , and for every Q ∈ Q and Q ∈ Q , there exists an ( β , α Q , α Q , ψ ( α Q , α Q ) , ψ ( α Q , α Q )) -blocker for ( j, t, | V ( T ) | + 1 , s + 2 , β − , t ′ ) .If either(i) α ′ α Q for every Q ∈ Q ∪ Q , or(ii) for every β ∈ [ α ′ − , ( t ′ , M ′ β ) is a nice pair,then there exists an ( α ′ − , α ′ , α ′ , ψ ( α ′ , α ′ ) , ψ ( α ′ , α ′ ) − ψ (0 , -blocker for ( j, t ′ , − , , , t ′ ) .Proof. We first prove that condition (ii) implies condition (i).Suppose that condition (ii) holds but condition (i) does not hold. Let β ∈ [ α ′ − .Let t ∗ ∈ V ( T ) be the node of T such that t ′ ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponentin G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ′ β ) and X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i t ∗ is minimum. Note that t is a candidate for t ∗ , so t ′ = t ∗ . Since V ( M ′ α ′ ) ∩ X t = ∅ , V ( M ′ β ) ∩ X t = ∅ . By Claim 4.12.31,there exists α (0) β ∈ [ β − such that M (0) α (0) β ⊆ M ′ β . So there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent R β in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] containing M (0) α (0) β . Since M (0) α (0) β inter-secting V ( M ′ β ) , by Claim 4.12.35, R β intersects X V ( T t ′ ) and satisfies A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( R β )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Since R β intersects X V ( T t ′ ) and V ( R β ) ∩ X t ⊇ V ( M (0) α (0) β ) ∩ X t = ∅ , R β intersects X t ′ . Since α (0) β < β , R β is a monochromatic E ( i t ∗ ,β − j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] . Since V ( R β ) ∩ X t ′ = ∅ and A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( R β )) ∩ X V ( T t ′ ) − X t ′ = ∅ ,by the maximality of β , there exists α β such that R β = M α β . Since α (0) β < β , α β < α Q forevery Q ∈ Q ∪ Q . Hence there exists a mapping ι that maps each element β ∈ [ α ′ − to α β . Note that ι is an injection. So there are at least α ′ − different numbers strictly smallerthan min Q ∈Q ∪Q α Q . Hence α ′ min Q ∈Q ∪ Q α Q and condition (i) holds, a contradiction.Therefore, to prove this claim, it suffices to assume the case that condition (i) holds.For every Q ∈ Q and Q ∈ Q , let κ Q ,Q = ψ ( α Q , α Q ) and ξ Q ,Q = ψ ( α Q , α Q ) , andlet t Q ,Q = ( t Q ,Q − ξ Q ,Q , . . . , t Q ,Q − , t Q ,Q , . . . , t Q ,Q κ Q ,Q ) be a ( β , α Q , α Q , κ Q ,Q , ξ Q ,Q ) -blockerfor ( j, t, | V ( T ) | +1 , s +2 , β − , t ′ ) such that P x i x is maximum, where the sum is over all entriesof t Q ,Q . We also denote κ Q ,Q by κ α Q ,α Q , and denote ξ Q ,Q by ξ α Q ,α Q . For every Q ∈ Q and Q ∈ Q , let α ∗ Q ,Q ∈ { α Q , α Q } such that A L ( it, | V ( T ) | +1 ,s +2) ( V ( M α ∗ Q ,Q )) ∩ ( X V ( T t ) − X t ) ∩ X V ( T q − ξQ ,Q ) − ( X q − ξQ ,Q ∪ I ◦ j ) = ∅ (if ξ Q ,Q = 0 ), and A L ( it, | V ( T ) | +1 ,s +2) ( V ( M α ∗ Q ,Q )) ∩ ( X V ( T t ) − X t ) ∩ X V ( T t ) − X t = ∅ , and subject to this, α ∗ Q ,Q = α Q if possible.77et M ′′ , M ′′ , . . . , M ′′ k (for some k ′′ ∈ N ) be the S ◦ j -related monochromatic E ( i t , | V ( G ) | ) j,t -pseudocomponents in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] such that for every α ∈ [ k ′′ ] , V ( M ′′ α ) ∩ X t ′ = ∅ and A L ( it ′ , − , ( V ( M ′ α )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Assume that σ ( M ′′ ) < σ ( M ′′ ) < · · · < σ ( M ′ k ′′ ) . Let V ( M ′′ γ ) = ∅ for every γ > k ′′ + 1 . Let b ′′ be the sequence ( | A L ( it, | V ( T ) | +1 ,s +2) ( V ( M ′′ )) ∩ X V ( T t ′ ) − ( X t ′ ∪ I ◦ j ) | , . . . , | A L ( it, | V ( T ) | +1 ,s +2) ( V ( M ′′ w )) ∩ X V ( T t ′ ) − ( X t ′ ∪ I ◦ j ) | ) .Let m = min Q ∈Q α Q . Let m = min Q ∈Q α Q .Note that by the choice of β , there exists no e ∈ E ( i t , | V ( G ) | ) j,t − E ( i t ,β − j,t such that e ∩ S m − γ =1 V ( M γ ) = ∅ . So either α ′ < m , or the subsequence of a ′ formed by the first α ′ − entries of a ′ is lexicographically at most the subsequence of a formed by the first m − entriesof a . That is, ψ ( α ′ , α ′ ) ψ ( m , m ) .We first assume that for every Q ∈ Q , α ∗ M m ,Q = α Q .Note that κ m ,m + ξ m ,m κ m ,α Q + ξ m ,α Q , and ( t M m ,M m ξ m ,m , . . . , t M m ,M m − , t M m ,M m , . . . , t M m ,M m κ m ,m ) is a suffix of ( t M m ,Q − ξ m ,αQ , . . . , t M m ,Q − , t M m ,Q , . . . , t M m ,Q κ m ,αQ ) for all Q ∈ Q by the choice of t M m ,Q .Since for every Q ∈ Q , α ∗ M m ,Q = α Q , we know S Q ∈Q A L ( it, | V ( T ) | +1 ,s +2) ( V ( Q )) ∩ ( X V ( T t ′ ) − X t ′ ) ∩ X V ( T tMm ,Mm ) − X t Mm ,Mm = ∅ and S Q ∈Q A L ( it, | V ( T ) | +1 ,s +2) ( V ( Q )) ∩ ( X V ( T t ′ ) − X t ′ ) ∩ X V ( T tMm ,Mm − ξm ,m ) − ( X t Mm ,Mm − ξm ,m ∪ I ◦ j ) = ∅ . Since for each vertex in ( Y ( i t ′ , − , − Y ( i t , | V ( T ) | +1 ,s +2) ) ∩ I j ∩ X V ( T t ′ ) , it is in X V ( T t ′ ) ∩ I j − ( X t ′ ∪ I ◦ j ) , andwhen it gets colored, it decreases the lexicographic order of b ′′ . So by Claim 4.12.24,we loss at most η · ( f ( η )) w − ( f ( η )) w ψ (0 , nodes in the blocker. Thatis, A L ( it ′ , − , ( V ( M ′ α ′ )) ∩ ( X V ( T t ′ ) − X t ′ ) ∩ ( X V ( T tMm ,Mm ) − X t Mm ,Mm ) = ∅ and A L ( it ′ , − , ( V ( M ′ α ′ )) ∩ ( X V ( T t ′ ) − X t ′ ) ∩ X V ( T tMm ,Mm − ξm ,m ψ , ) − ( X t Mm ,Mm − ξm ,m ψ , ∪ I ◦ j ) = ∅ .So ( t M m ,M m − ξ m ,m + ψ (0 , , . . . , t M m ,M m − , t M m ,M m , . . . , t M m ,M m κ m ,m ) contains a subsequence that is an ( α ′ − , α ′ , α ′ , ψ ( α ′ , α ′ ) , ψ ( α ′ , α ′ ) − ψ (0 , -blocker for ( j, t ′ , − , , , t ′ ) .Hence we may assume that there exists Q ∗ ∈ Q such that m = α ∗ M m ,Q ∗ = α Q ∗ . Since forevery Q ∈ Q − { M m } , ψ ( α Q , α Q ∗ ) > ψ ( m , α Q ∗ ) + ψ ( m , α Q ∗ ) , we know α ∗ Q ,Q ∗ = α Q ,for otherwise α ∗ M m ,Q ∗ = α Q ∗ . So ( t M m ,Q ∗ − ξ m ,αQ ∗ , . . . , t M m ,Q ∗ − , t M m ,Q ∗ , . . . , t M m ,Q ∗ κ m ,αQ ∗ ) is a suffix of ( t Q ,Q ∗ − ξ αQ ,αQ ∗ , . . . , t Q ,Q ∗ − , t Q ,Q ∗ , . . . , t Q ,Q ∗ κ αQ ,αQ ∗ ) for every Q ∈ Q . Note that for each vertex in ( Y ( i t ′ , − , − Y ( i t , | V ( T ) | +1 ,s +2) ) ∩ I j ∩ X V ( T t ′ ) , it is in X V ( T t ′ ) − ( X t ′ ∪ I ◦ j ) , and when it getscolored, it decreases the lexicographic order of b ′′ . So by Claim 4.12.24, we loss at most η · ( f ( η )) w − ( f ( η )) w ψ (0 , nodes in the blocker. Hence ( t M m ,Q ∗ − ξ m ,αQ ∗ + ψ (0 , , . . . , t M m ,Q ∗ κ m ,αQ ∗ ) contains a subsequence that is an ( α ′ − , α ′ , α ′ , ψ ( α ′ , α ′ ) , ψ ( α ′ , α ′ ) − ψ (0 , -blocker for ( j, t ′ , − , , , t ′ ) . This proves the claim. Claim 4.12.41. Let j ∈ [ |V| − . Let t ∈ V ( T ) . Let M , M , . . . , M k (for some k ∈ N ) be the S ◦ j -related monochromatic E ( i t ) j,t -pseudocomponents in G [ Y ( i t , − , ] such that for every α ∈ [ k ] , V ( M α ) ∩ X t = ∅ and A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t = ∅ . Assume that σ ( M ) < σ ( M ) < · · · < σ ( M k ) . Let V ( M γ ) = ∅ for every γ > k + 1 . Let α , α ∈ [2 , k ] with α < α .Assume that for every α ∈ [2 , α ] and α ′ ∈ [ α + 1 , k ] , if either α α − or ( α, α ′ ) = ( α , α ) , • M α and M α ′ have the same color, and • the s -segment in S ◦ j containing V ( M α ) whose level equals the color of M α contains V ( M α ′ ) ,then either • there exists an ( α − , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ ) − ψ (0 , -blocker for ( j, t, − , , , t ) , or • there exist t ∗ ∈ V ( T ) with i t ∗ < i t and t ∈ V ( T j,t ∗ ) , a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M α ) ∪ V ( M α ′ ) .Assume ( z, M z ) is a nice pair and an outer-safe pair for every z ∈ V ( T ) and monochromatic E ( i z ) j,z -pseudocomponent M z in G [ Y ( i z , − , ] with V ( M z ) ∩ X z = ∅ and σ ( M z ) < σ ( M α ) .Then either • there exists no c -monochromatic path P in G from V ( M α ) ∩ V ( P ) to V ( M α ) ∩ V ( P ) internally disjoint from S w γ =1 V ( M γ ) such that V ( P ) ∩ A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t = ∅ , or • ( t, M α ) is not an outer-safe pair.Proof. Suppose this claim does not hold. Among all counterexamples, we choose ( t, M α ) such that σ ( M α ) is as small as possible, and subject to this, i t is as large as possible.Hence ( t, M α ) is an outer-safe pair, and there exists a c -monochromatic path P in G from V ( M α ) ∩ V ( P ) to V ( M α ) ∩ V ( P ) internally disjoint from S w γ =1 V ( M γ ) such that V ( P ) ∩ A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t = ∅ .Since for every α ∈ [2 , α − , ( t, M α ) is an outer-safe pair, applying this claim to M α implies that for every α ∈ [2 , α − and α ′ ∈ [ α + 1 , k ] , there exists no c -monochromaticpath P in G from V ( M α ) ∩ V ( P ) to V ( M α ′ ) ∩ V ( P ) internally disjoint from S w γ =1 V ( M γ ) suchthat V ( P ) ∩ A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t = ∅ . Note that the case α = 1 is also true byClaim 4.12.34. So we have(a) for every α ∈ [ α − and α ′ ∈ [ α + 1 , k ] , there exists no c -monochromatic path P in G from V ( M α ) ∩ V ( P ) to V ( M α ′ ) ∩ V ( P ) internally disjoint from S w γ =1 V ( M γ ) such that V ( P ) ∩ A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t = ∅ .Suppose there exist α ∈ [2 , α ] and α ′ ∈ [ α + 1 , k ] with either α α − or ( α, α ′ ) = ( α , α ) , such that M α and M α ′ have the same color, and the s -segment in S ◦ j containing V ( M α ) whose level equals the color of M α contains V ( M α ′ ) , and there exists no ( α − , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ ) − ψ (0 , -blocker for ( j, t, − , , , t ) , and for every t ∗ ∈ V ( T ) with i t ∗ < i t and t ∈ V ( T j,t ∗ ) , if there exist a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersect-ing X q ′ and V ( M α ) ∪ V ( M α ′ ) , then t ∈ ∂T j,t ∗ . By our assumption, such t ∗ exists, and we choose t ∗ such that i t ∗ is as small as possible. So t ∈ ∂T j,t ∗ . By Claims 4.12.25 and 4.12.33, t is a witessfor X t ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] . Let t ∗ be the node of T with t ∈ V ( T j,t ∗ ) suchthat some monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and79 ( M ) for some witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and subjectto this, i t ∗ is minimum. Since V ( M ) ∩ X t = ∅ , t ∗ is a candidate of t ∗ . So i t ∗ i t ∗ and hence t ∗ = t . Since ( t, M ) is a nice pair, by Claim 4.12.36, there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) such that σ ( M ∗ ) = σ ( M ) , V ( M ∗ ) ∩ X t = ∅ and satisfies A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X V ( T t ) − X t = ∅ . By the minimalityof t ∗ and the construction of E ( i t ∗ , | V ( G ) | ) j,t ∗ , since i t ∗ i t ∗ and α = α ′ , V ( M α ) ∩ X t ∗ = ∅ andthere exist blockers for ( j, t ∗ , | V ( T ) | +1 , s +2 , β, t ) (for some β ∈ [0 , | V ( G ) | ] ) as stated in the as-sumption of Claim 4.12.40. Since α α , the assumption of this claim implies that condition(ii) in Claim 4.12.40 holds. So there is an ( α − , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ ) − ψ (0 , -blockerfor ( j, t, − , , , t ) by Claim 4.12.40, a contradiction.Therefore, we have(b) for every α ∈ [2 , α ] and α ′ ∈ [ α + 1 , k ] , if – either α α − or ( α, α ′ ) = ( α , α ) , – M α and M α ′ have the same color, and – the s -segment in S ◦ j containing V ( M α ) whose level equals the color of M α contains V ( M α ′ ) ,then there exists no c -monochromatic path P in G from V ( M α ) ∩ V ( P ) to V ( M α ′ ) ∩ V ( P ) internally disjoint from S w γ =1 V ( M γ ) such that V ( P ) ∩ A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t = ∅ , and either – there exists an ( α − , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ ) − ψ (0 , -blocker for ( j, t, − , , , t ) , or – there exist t ∗ ∈ V ( T ) with i t ∗ < i t and t ∈ V ( T j,t ∗ ) − ∂T j,t ∗ , a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M α ) ∪ V ( M α ′ ) .Now we prove that(c) there exist no t ∗ ∈ V ( T ) with i t ∗ < i t and t ∈ V ( T j,t ∗ ) − ∂T j,t ∗ , a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M α ) ∪ V ( M α ) .Suppose to the contrary that (c) does not hold. So there exist t ∗ ∈ V ( T ) with i t ∗ < i t and t ∈ V ( T j,t ∗ ) − ∂T j,t ∗ , a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] ,and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M α ) ∪ V ( M α ) . We choose t ∗ such that i t ∗ is as small as possible. For each γ ∈ [2] , let t ∗ α γ bethe node of T with t ∈ V ( T j,t ∗ αγ ) such that some monochromatic E ( i t ∗ αγ ) j,t ∗ αγ -pseudocomponent in G [ Y ( i t ∗ αγ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M α γ ) for some witness q ′ ∈ ∂T j,t ∗ αγ for X q ′ ∩ I j ⊆ W ( i t ∗ αγ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and subject to this, i t ∗ αγ is minimum. Note that t ∗ isa candidate for t ∗ α γ for some γ ∈ [2] . So i t ∗ > min { i t ∗ α , i t ∗ α } . Let t ∈ { t ∗ α , t ∗ α } with i t = min { i t ∗ α , i t ∗ α } . If i t < i t ∗ , then since t ∈ V ( T j,t ∗ ) − ∂T j,t ∗ , t ∈ V ( T j,t ) − ∂T j,t , so t 80s a candidate for t ∗ . Hence t ∗ = t ∗ α γ for some γ ∈ [2] . By Claims 4.12.25 and 4.12.33 andby tracing along P , we have t ∗ = t ∗ α = t ∗ α . Since α = α , by Claim 4.12.39, there exist t ′ ∈ ∂T j,t ∗ α ∩ V ( T t ) , u, v ∈ X t ′ , a c -monochromatic path P ′ in G [ X V ( T t ′ ) ] but not in G [ Y ( i t , − , ] from u to v internally disjoint from X t ′ such that t ′ is a witness for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) = σ ( M α ) containing u but not containing v . Since t ∈ V ( T j,t ∗ ) − ∂T j,t ∗ , t ′ ∈ V ( T t ) − { t } . Let M be the monochromatic E ( i t ′ ) j,t ′ -pseudocomponentin G [ Y ( i t ′ , − , ] containing M ∗ . Since t ′ ∈ ∂T j,t ∗ , by Claims 4.12.25 and 4.12.33, t ∗ is the nodeof T with t ′ ∈ V ( T j,t ∗ ) such that there exist a witness q ′′ ∈ ∂T j,t ∗ for X q ′′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′′ and V ( M ) , and subject to this, i t ∗ is minimum.Hence there exist q ∈ ∂T j,q ∗ ∩ V ( T t ) − { t } , a monochromatic E ( i q ) j,q -pseudocomponent Q in G [ Y ( i q , − , ] , u q , v q ∈ X q , a c -monochromatic path P q in G [ X V ( T q ) ] but not in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] from u q to v q internally disjoint from X q , and a monochromatic E ( i q ∗ , | V ( G ) | ) j,q ∗ -pseudocomponent Q ∗ in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] with σ ( Q ∗ ) σ ( M α ) containing u q but not containing v q such that Q ⊇ Q ∗ , T j,t ∗ ⊆ T j,q ∗ and q is a witenss for X q ∩ I j ⊆ W ( i q ∗ ,ℓ ′ )3 forsome ℓ ′ ∈ [0 , | V ( T ) | ] , where q ∗ is the node with i q ∗ < i q and q ∈ V ( T j,q ∗ ) such that there exist awitness q ′′ ∈ ∂T j,q ∗ for X q ′′ ∩ I j ⊆ W ( i q ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i q ∗ ) j,q ∗ -pseudocomponent in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′′ and V ( Q ) , and subject to this, i q ∗ is minimum. Note that such ( q, q ∗ , Q, u q , v q , P q , Q ∗ ) exists since ( t ′ , t ∗ , M , u, v, P t ′ , M ∗ ) is acandidate for ( q, q ∗ , Q, u q , v q , P q ′ , Q ∗ ) . We further choose them such that σ ( Q ∗ ) is as small aspossible. Note that σ ( Q ) σ ( Q ∗ ) σ ( M α ) .Suppose V ( P q ) ⊆ Y ( i q , − , . Since σ ( Q ) σ ( M α ) , we can apply Claim 4.12.37 by taking ( t, M, t ′ ) to be ( q, Q, q ) . By applying Claim 4.12.37 by taking ( t, M, t ′ ) to be ( q, Q, q ) , we know σ ( Q ∗ ) > σ ( Q ) . So by Claims 4.12.25, 4.12.31 and 4.12.33, there exists a monochromatic E ( i q ∗ ) j,q ∗ -pseudocomponent Q ′′ in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] with σ ( Q ′′ ) = σ ( Q ) < σ ( Q ∗ ) . Since Q ′′ and Q ∗ are disjoint subgraphs of Q , by Claims 4.12.25 and 4.12.33, there exist q ′′ ∈ ∂T j,q ∗ , u q ′′ , v q ′′ ∈ X q ′′ , a monochromatic path P q ′′ in G [ Y ( i q , − , ∩ X V ( T q ′′ ) ] but not in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] from u q ′′ to v q ′′ internally disjoint from X q ′′ such that Q ′′ contains u q ′′ but not v q ′′ . Since σ ( Q ) <σ ( Q ∗ ) σ ( M α ) , ( q, Q ) is outer-safe, so q ′′ ∈ V ( T q ) . Since q ∈ ∂T j,q ∗ , q ′′ = q . Since σ ( Q ′′ ) < σ ( Q ∗ ) , Q ′′ contradicts the choice of Q ∗ .So V ( P q ) Y ( i q , − , . Hence, by Claims 4.12.34 and 4.12.36 and the assumption of thisclaim, there exists a monochromatic E ( i q ∗ , | V ( G ) | ) j,q ∗ -pseudocomponent Q ∗ in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] such that σ ( Q ∗ ) < σ ( Q ∗ ) , V ( Q ∗ ) ∩ X q = ∅ and A L ( iq ∗ , | V ( T ) | +1 ,s +2) ( V ( Q ∗ )) ∩ X V ( T q ) − X q = ∅ . Let R ∗ be the monochromatic E ( i q ) j,q -pseudocomponent in G [ Y ( i q , − , ] containing Q ∗ .Let R ∗ , . . . , R ∗ β (for some β ∈ N ) be the monochromatic E ( i q ) j,q -pseudocomponents M ′ in G [ Y ( i q , − , ] intersecting X q with A L ( iq, − , ( V ( M ′ )) ∩ X V ( T q ) − X q = ∅ and σ ( M ′ ) < σ ( R ∗ ) .Assume σ ( R ∗ ) < σ ( R ∗ ) < · · · < σ ( R ∗ β ) . Let R ∗ β +1 = R ∗ . Since u q ∈ V ( Q ∗ ) and v q V ( Q ∗ ) , { u q , v q } 6∈ E ( i q ∗ , | V ( G ) | ) j,q ∗ . By the existence of Q ∗ and the construction of E ( i q ∗ , | V ( G ) | ) j,q ∗ , V ( Q ∗ ) ∩ X q ∗ = ∅ . Since σ ( Q ∗ ) σ ( M α ) , for every α ∈ [ β ] , R ∗ α contains M α for some α ∈ [ α − by Claim 4.12.31. So by the assumption of this claim and Claim 4.12.40, the81ssumptions of this claim hold when t is replaced by q and M α is replaced by R ∗ β +1 = R ∗ .Since σ ( R ∗ ) σ ( M α ) and i q > i t , this claim is applicable to q and R ∗ , and the exis-tence of P q implies that ( q, R ∗ ) is not an outer-safe pair. So σ ( R ∗ ) = σ ( M α ) > σ ( Q ∗ ) .So σ ( R ∗ ) = σ ( Q ∗ ) and R ∗ = Q ⊇ M α . Hence t ∗ = q ∗ by Claims 4.12.25 and 4.12.33.Since ( q, R ∗ ) is not an outer-safe pair, there exist t ′′ ∈ ∂T j,q ∗ − V ( T q ) , u ′′ , v ′′ ∈ X t ′′ , amonochromatic path P ′′ in G [ Y ( i q , − , ∩ X V ( T t ′′ ) ] but not in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] from u ′′ to v ′′ internally disjoint from X t ′′ , and a monochromatic E ( q ∗ , | V ( G ) | ) j,q ∗ -pseudocomponent M ′′ in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ′′ ) = σ ( R ∗ ) = σ ( M α ) containing u ′′ but not v ′′ . Since q ∈ V ( T t ) and V ( P ′′ ) ⊆ Y ( i q , − , ∩ X V ( T t ′′ ) , if t ′′ V ( T t ) − { t } , then V ( P ′′ ) ⊆ Y ( i t , − , , so it contradictsthat ( t, M α ) is an outer-safe pair. So t ′′ ∈ ∂T j,q ∗ ∩ ( V ( T t ) − { t } ) − V ( T q ) . In particular, t ′′ = q .If V ( P ′′ ) Y ( i t ′′ , − , , then by Claims 4.12.34 and 4.12.36, the fact that { u ′′ , v ′′ } 6∈ E ( i q ∗ , | V ( G ) | ) j,q ∗ , the existence of P q and P t ′′ , and the construction of E ( i q ∗ , | V ( G ) | ) j,q ∗ , we know A L ( iq ∗ , | V ( T ) | +1 ,s +2) ( V ( Q ∗ )) ∩ X V ( T q ∗ ) − X q ∗ ⊆ ( X V ( T q ) − X q ) ∩ ( X V ( T t ′′ ) − X t ′′ ) , so t ′′ = q , acontradiction. Hence V ( P ′′ ) ⊆ Y ( i t ′′ , − , . Since σ ( M ′′ ) = σ ( R ∗ ) = σ ( Q ∗ ) , M ′′ = Q ∗ . Let M ′′′ be the monochromatic E ( i t ′′ ) j,t ′′ -pseudocomponent in G [ Y ( i t ′′ , − , ] containing M ′′ . Since σ ( M ′′ ) = σ ( Q ∗ ) = σ ( M α ) , M ′′′ contains M α , so t ∗ = q ∗ is the node in T with t ′′ ∈ ∂T j,t ∗ such that there exist a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] and amonochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M ′′′ ) .Since σ ( M ′′ ) = σ ( Q ∗ ) = σ ( M α ) , we can apply Claim 4.12.37 by taking ( t, M, t ′ ) tobe ( t ′′ , M ′′′ , t ′′ ) . By applying Claim 4.12.37 by taking ( t, M, t ′ ) to be ( t ′′ , M ′′′ , t ′′ ) , we know σ ( Q ∗ ) = σ ( M ′′ ) > σ ( M ′′′ ) . So by Claims 4.12.25, 4.12.31 and 4.12.33, there exists a monochro-matic E ( i q ∗ , | V ( G ) | ) j,q ∗ -pseudocomponent Q ′′′ in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] with σ ( Q ′′′ ) = σ ( M ′′′ ) <σ ( M ′′ ) = σ ( Q ∗ ) . Hence there exist q ′′′ ∈ ∂T j,q ∗ , u q ′′′ , v q ′′′ ∈ X q ′′′ , a monochromatic path P q ′′′ in G [ Y ( i t ′′ , − , ∩ X V ( T q ′′′ ) ] but not in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] from u q ′′′ to v q ′′′ internally dis-joint from X q ′′′ such that Q ′′′ contains u q ′′′ but not v q ′′′ . Since σ ( Q ′′′ ) < σ ( Q ∗ ) σ ( M α ) , ( t ′′ , M ′′′ ) is outer-safe, so q ′′′ ∈ V ( T t ′′ ) . Hence q ′′′ = t ′′ . Since σ ( Q ′′′ ) < σ ( Q ∗ ) , Q ′′′ contradictsthe choice of Q ∗ .This proves (c).By the existence of P , M α and M α have the same color, and the s -segment in S ◦ j containing V ( M α ) whose level equals the color of M α contains V ( M α ) . By (b) and (c),there exists an ( α − , α , α , κ, ξ ) -blocker ( t − ξ , t − ξ +1 , . . . , t − , t , t , . . . , t κ ) for ( j, t, − , , , t ) ,where κ = ψ ( α , α ) and ξ = ψ ( α , α ) − ψ (0 , . So S α − α =1 A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t ⊆ ( X V ( T tκ ) − X t κ ) ∩ I ◦ j . Let α ∗ be the element in { α , α } such that A L ( it, − , ( V ( M α ∗ )) ∩ ( X V ( T t ) − ( X t ∪ I ◦ j )) ∩ X V ( T t − ξ ) − X t − ξ = ∅ (if ξ = 0 ), and A L ( it, − , ( V ( M α ∗ )) ∩ ( X V ( T t ) − X t ) ∩ X V ( T t ) − X t = ∅ .Now we prove(d) t η V ( T j,t ) .Suppose to the contrary that t η ∈ V ( T j,t ) .Suppose ( Y ( i t , − ,α − − Y ( i t , − , ) X V ( T tκ − α η ) . By Claim 4.12.24, since S α − α =1 A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t ⊆ ( X V ( T tκ ) − X t κ ) ∩ I ◦ j and { X t γ − X t : γ ∈ [ − ξ, κ ] −{ }} consists of pariwise disjoint members, there exist α ∈ [ α − , α ′ ∈ [ α + 1 , w ] and a82onochromatic path P ′ in G [ Y ( i t , − ,α − ] from V ( M α ) to V ( M α ′ ) internally disjoint from S w γ =1 V ( M γ ) such that all internal vertices of P ′ are in Y ( i t , − , ∪ ( I ◦ j ∩ X V ( T tκ − α η ) − X t κ − α η ) and V ( P ′ ) ∩ A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t = ∅ , a contradiction.Hence ( Y ( i t , − ,α − − Y ( i t , − , ) ⊆ X V ( T tκ − α η ) . Since A L ( it, − , ( V ( M α ∗ )) ∩ ( X V ( T t ) − X t ) ∩ ( X V ( T t ) − X t ) = ∅ and κ − α η > , M α ∗ is a monochromatic E ( i t ) j,t -pseudocomponentin G [ Y ( i t , − ,α − ] . So V ( P ) Y ( i t , − ,α − . Let M ′ be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − ,α − ] containing M α . Since V ( P ) Y ( i t , − ,α − , there exists u ∈ V ( P ) ∩ A L ( it ∗ , − ,α − ( V ( M ′ )) ∩ X V ( T t ) − X t . Then either u is colored in Y ( i t , − ,α ) by acolor different from c ( P ) , or u Y ( i t , | V ( T ) | +1 ,s +2) . The former cannot hold, so the latter holds.So u Z t . Hence there exists q u ∈ ∂T j,t with u ∈ I ◦ j ∩ X V ( T qu ) − X q u .Let M ′′ be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] containing M α ∗ . Since u Y ( i t , | V ( T ) | +1 ,s +2) , there exists v ∈ V ( P ) ∩ A L ( it, | V ( T ) | +1 ,s +2) ( V ( M ′′ )) ∩ X V ( T t ) − X t .We choose v such that σ ( v ) is as small as possible. Since P is internally disjoint from X t and A L ( it, − , ( M α ∗ ) ∩ X V ( T t ) − X t ⊆ V ( G ) − ( X V ( T t ) − X t ) , by Claim 4.12.24, v ∈ V ( G ) − X V ( T tη ) .Note that v Z t . Hence there exists q v ∈ ∂T j,t with v ∈ X V ( T qv ) − X q v .By Claim 4.12.24, | Y ( i t , | V ( T ) | +1 ,s +2) ∩ I j | η . Since t η ∈ V ( T j,t ) , there exist η + 1 β <β η − α η with β − β > η − ( α +1) η η +1 > η such that Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T tβ ) − X V ( T tβ ) = ∅ . Since v ∈ V ( G ) − X V ( T tη ) , q v ∈ V ( T ) − V ( T t β ) .Suppose that q u ∈ V ( T t β ) . So α ∗ = α . Since Y ( i t , | V ( T ) | +1 ,s +2) ∩ X V ( T tβ ) − X V ( T tβ ) = ∅ , q u ∈ V ( T t β ) . By the existence of P and Claims 4.12.25 and 4.12.33, there exist subpathsof P contained in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] such that for each β ∈ [ β , β ] , there exists a subpathintersecting X t β − X t . But it contradicts Claim 4.12.24 since β − β > η .Hence q u ∈ V ( T ) − V ( T t β ) . Let M ∗ be the monochromatic E ( i qu ) j,q u -pseudocomponent in G [ Y ( i qu , − , ] containing M α . Note that u ∈ V ( P ) ∩ A L ( iqu, − , ( V ( M ∗ )) ∩ X V ( T qu ) − X q u since u ∈ I ◦ j ∩ X V ( T qu ) − X q u . By Claim 4.12.34, M ∗ does not have the smallest σ -value among all S ◦ j -related monochromatic E ( i qu ) j,q u -pseudocomponents M ′′′ in G [ Y ( i qu , − , ] intersecting X q u with A L ( iqu, − , ( V ( M ′′′ )) ∩ X V ( T qu ) − X q u = ∅ . So there exists an S ◦ j -related monochromatic E ( i qu ) j,q u -pseudocomponents Q ∗ in G [ Y ( i qu , − , ] intersecting X q u with A L ( iqu , − , ( V ( M ′′′ )) ∩ X V ( T qu ) − X q u = ∅ such that σ ( Q ∗ ) < σ ( M ∗ ) . By Claim 4.12.31, there exists α ∈ [ α − such that M α ⊆ Q ∗ . By Claims 4.12.25 and 4.12.33, there exists a monochromatic E ( i t , | V ( G ) | ) j,t -pseudocomponent Q in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] intersecting X q u and containing M α . Since Q contains M α , and Y ( i t , − ,α − − Y ( i t , − , ⊆ X V ( T tκ − α η ) , by (a) and Claim 4.12.24, V ( Q ) − X V ( T tκ − α η − η ) = ∅ .But T t κ − α η ⊆ T t β and β − β > η , V ( Q ) ∩ X q u = ∅ , a contradiction.This proves that t η V ( T j,t ) .So there exists t ′ ∈ ∂T j,t such that t η ∈ V ( T t ′ ) − { t ′ } . Since S α − α =1 A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t ⊆ ( X V ( T tκ ) − X t κ ) ∩ I ◦ j , M α is a monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] and V ( P ) ∩ A L ( it, | V ( T ) | +1 ,s +2) ( V ( M α )) ∩ X V ( T t ) − X t is a nonempty subsetof I ◦ j .Since the process for adding fake edges in one interface is irrelevant with the processfor adding fake edges in other interfaces, to simplify the notation, we may without loss of83enerality assume that for every ℓ ∈ [0 , | V ( G ) | ] , the monochromatic E ( i t ,ℓ ) j,t -pseudocomponent M ′ in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] in which σ ( M ′ ) is the ( ℓ + 1) -th smallest among all monochromatic E ( i t ,ℓ ) j,t -pseudocomponents in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] is S ◦ j -related, unless there are less than ℓ + 1 S ◦ j -related monochromatic E ( i t ,ℓ ) j,t -pseudocomponents in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] .For every γ ∈ [ w ] and γ ′ ∈ [0 , | V ( G ) | ] , let M ( γ ′ ) γ be the S ◦ j -related monochromatic E ( i t ,γ ′ ) j,t -pseudocomponent in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] with the γ -th smallest σ -value among all S ◦ j -relatedmonochromatic E ( i t ,γ ′ ) j,t -pseudocomponents M ′ in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] with V ( M ′ ) ∩ X t ′ = ∅ and A L ( it, | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ′ ) − X t ′ = ∅ (if there are at most γ − such M ′ , then let M ( γ ′ ) γ = ∅ ). By Claim 4.12.31, each M ( γ ′ ) γ either contains M γ ′′ for some γ ′′ , or is disjoint from X t . We shall prove the following statements (i) and (ii) by induction on α : for every α ∈ [0 , α − and α ∈ [ α + 1 , α − ,(i) there exists no e ∈ E ( i t ,α ) j,t − E ( i t ) j,t such that e ∩ V ( M ( α ) α ) = ∅ , and(ii) for every α ′ ∈ [ α + 1 , w ] , if α > , and M ( α − α and M ( α − α ′ have the same color,and the s -segment in S ◦ j containing V ( M ( α − α ) whose level equals the color of M ( α − α contains V ( M ( α − α ′ ) , then either – there exists an ( α , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ )) -blocker for ( j, t, | V ( T ) | + 1 , s + 2 , α − , t ′ ) , or – there exist t ∗ ∈ V ( T ) with i t ∗ < i t ′ and t ′ ∈ V ( T j,t ∗ ) , a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M ( α − α ) ∪ V ( M ( α − α ′ ) .Note that (i) and (ii) obviously hold for α = 0 . And for every α > , (ii) for α implies (i)for α by the construction for E ( i t ,α ) j,t and Claims 4.12.25 and 4.12.33. So it suffices to showthat if (i) holds for every smaller α , then (ii) holds for α .Assume (i) holds for every smaller α . That is, there exists no e ∈ E ( i t ,α − j,t − E ( i t ) j,t and α ′′ ∈ [0 , α − such that e ∩ V ( M ( α − α ′′ ) = ∅ . So M ( α − γ = M γ for every γ ∈ [ α − . Since α α − , ψ ( α, α ′ )+ ψ ( α, α ′ ) ψ ( α , α ) − f ( η ) w +3 . If V ( M ( α − α ′ ) ∩ X t = ∅ , then since t ( f ( η )) w ∈ V ( T t ′ ) and ( t , t , . . . , t κ ) is an ( α − , α , α , κ, -blocker for ( j, t, − , , , t ) ,we know A L ( it, | V ( T ) | +1 ,s +2) ( V ( M ( α − α ′ )) ∩ X V ( T t ( f η w η ) = ∅ by Claim 4.12.24, so ( t ( f ( η )) w +3 η +1 , . . . , t κ ) contains a subsequence that is an ( α , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ )) -blocker for ( j, t, | V ( T ) | + 1 , s + 2 , α − , t ′ ) . So we may assume that V ( M ( α − α ′ ) ∩ X t = ∅ .Hence M ( α − α ′ contains M α ′′ for some α ′′ ∈ [ α ′ , k ] by Claim 4.12.31. Let S = { β ∈ [ w ] : M β ⊆ M ( α − α ′ } . Note that for every β ∈ S , β > α ′ > α + 1 .If there exist β ∈ S and t ∗ ∈ V ( T ) with i t ∗ < i t and t ∈ V ( T j,t ∗ ) − ∂T j,t ∗ , awitness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochro-matic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M α ) ∪ V ( M β ) ,then i t ∗ < i t ′ and t ′ ∈ V ( T j,t ∗ ) , so the second case for (ii) holds and we are done.84o by (b), we may assume that fore every β ∈ S , there exists an ( α , α, β, κ β , ξ β ) -blocker q β = ( q β − ξ β , q β − ξ β +1 , . . . , q β − , q β , q β , . . . , q βκ β ) for ( j, t, − , , , t ) , where κ β = ψ ( α, β ) and ξ β = ψ ( α, β ) − ψ (0 , . We further choose each q β such that the sequence ( i q β − ξβ , i q β − ξβ +1 , . . . , i q β − , i q β , i q β , . . . , i q βκβ ) is lexicographically maximal.Since β > α ′ > α + 1 for every β ∈ S , we know κ β is identical for all β ∈ S , and ξ β isidentical for all β ∈ S . Since q β are chosen such that the sequence of the heights of the nodesin q β is lexicographically maximal, q β is identical for all β ∈ S .Since α α − , for every β , κ β + ξ β ψ ( α , α ) − (2 f ( η ) w +3 +3 η + ψ (20 , κ − (2 f ( η ) w +3 + 3 η + ψ (20 , , so if t f ( η ) w +3 η + ψ (20 , ∈ V ( T q β ) , then ( q β − ξ β , . . . , q β − , t f ( η ) w +4 η + ψ (20 , , . . . , t κ ) contains a subsequence thatis an ( α , α, β, κ β , ξ β ) -blocker for ( j, t, − , , , t ) , contradicting the maximality of theheights of the nodes. Hence for every β ∈ S , q β ∈ V ( T t f η w η ψ , ) −{ t f ( η ) w +3 η + ψ (20 , } . Note that for every β ∈ S , ψ ( α, β ) ψ (3 , .So if there exists β ∈ S such that t f ( η ) w +3 η + ψ (10 , ∈ V ( T q β − ξβ ) , then ( t f ( η ) w +3 η + ψ (10 , , . . . , t f ( η ) w +3 η + ψ (20 , , q β , . . . , q βκ β ) contains a subsequence thatis an ( α , α, β, κ β , ξ β ) -blocker for ( j, t, − , , , t ) , contradicting the lexicographical maximal-ity. Hence for every β ∈ S , q β − ξ β ∈ V ( T t f η w η ψ , ) − { t f ( η ) w +3 η + ψ (10 , } .Note that for each v ∈ S w γ =0 V ( M ( α − γ ) , if there exists a monochromatic path in G [ S w γ =0 V ( M ( α − γ )] + E ( i t ) j,t from v to X t , then A L ( it, | V ( T ) | +1 ,s +2) ( { v } ) ∩ X V ( T t ′ ) − ( X t ′ ∪ I ◦ j ) = ∅ ; if A L ( it, | V ( T ) | +1 ,s +2) ( { v } ) ∩ X V ( T t ′ ) − ( X t ′ ∪ I ◦ j ) = ∅ and there exists no monochro-matic path in G [ S w γ =0 V ( M ( α − γ )] + E ( i t ) j,t from v to X t , then since t η ∈ V ( T t ′ ) and ( t , t , . . . , t κ ) is a blocker for ( j, t, − , , , t ) , we know v ∈ V ( G ) − ( X V ( T tη η ) ∪ X t ) and A L ( it, | V ( T ) | +1 ,s +2) ( { v } ) ∩ X V ( T t ′ ) − ( X t ′ ∪ I ◦ j ) ⊆ V ( G ) − X V ( T tη η · f η w ) by Claim 4.12.24. Hence A L ( it, | V ( T ) | +1 ,s +2) ( S w γ =0 V ( M ( α − γ )) ∩ X V ( T t ′ ) − ( X t ′ ∪ I ◦ j ) ⊆ V ( G ) − X V ( T tη η · f η w ) .For every β ∈ S , let α ∗ β ∈ { α, β } such that A L ( it, − , ( M α ∗ β ) ∩ ( X V ( T t ) − X t ) ∩ X V ( T qβ ) − X q β = ∅ and A L ( it, − , ( M α ∗ β ) ∩ ( X V ( T t ) − X t ) ∩ X V ( T qβ − ξβ ) − ( X q β − ξβ ∪ I ◦ j ) = ∅ .So for every β ∈ S , ( t f ( η ) w +3 η + ψ (5 , , . . . , t f ( η ) w +3 η + ψ (10 , , q β − ξ β , . . . , q βκ β ) is an ( α , α, β, ψ ( α, β ) , ψ ( α, β ) − ψ (0 , 0) + ψ (5 , -blocker for ( j, t, | V ( T ) | + 1 , s + 2 , t ′ ) . Notethat for every β ∈ S , β > α ′ and ψ ( α, β ) − ψ (0 , 0) + ψ (5 , > ψ ( α, α ′ ) . Hence, if α ∗ β = α for some β ∈ S , then ( t f ( η ) w +3 η + ψ (5 , , . . . , t f ( η ) w +3 η + ψ (10 , , q β − ξ β , . . . , q βκ β ) contains a subsequence that is an ( α , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ )) -blocker for ( j, t, | V ( T ) | +1 , s + 2 , t ′ ) , so we are done. So we may assume that α ∗ β = β for every β ∈ S . Then ( t f ( η ) w +3 η + ψ (5 , , . . . , t f ( η ) w +3 η + ψ (10 , , q β − ξ β , . . . , q βκ β ) contains a subsequence thatis an ( α , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ )) -blocker for ( j, t, | V ( T ) | + 1 , s + 2 , t ′ ) , so we are done.Therefore, (i) and (ii) hold for all α ∈ [0 , α − . In particular, there exists no e ∈ E ( i t ,α − j,t − E ( i t ) j,t and α ′′ ∈ [ α − such that e ∩ V ( M ( α − α ′′ ) = ∅ . Hence there exists no e ∈ E ( i t , | V ( G ) | ) j,t − E ( i t ) j,t such that e ∩ S α − γ =1 V ( M γ ) = ∅ . So S M ′ A L ( it, | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ) − X t ⊆ S α − α =1 A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t ⊆ X V ( T tκ ) ∩ I ◦ j , where the first unionis over all S ◦ j -related monochromatic E ( i t , | V ( G ) | ) j,t -pseudocomponents M ′ in G [ Y ( i t , | V ( T ) | +1 ,s +2) ] σ ( M ′ ) < σ ( M α ) .For each γ ∈ [ w ] , let M ′′ γ be the S ◦ j -related monochromatic E ( i t ′ ) j,t ′ -pseudocomponentin G [ Y ( i t ′ , − , ] with the γ -th smallest σ -value among all S ◦ j -related monochromatic E ( i t ′ ) j,t ′ -pseudocomponents M ′ in G [ Y ( i t ′ , − , ] with V ( M ′ ) ∩ X t ′ = ∅ and A L ( it ′ , − , ( V ( M ′ )) ∩ X V ( T t ′ ) − X t ′ = ∅ (if there are less than γ such M ′ , then let V ( M ′′ γ ) = ∅ ).Since (i) and (ii) hold for α ∈ [0 , α − , by Claim 4.12.40, for every α ∈ [2 , α − and α ′ ∈ [ α + 1 , w ] , if M ′′ α and M ′′ α ′ have the same color, and the s -segment in S ◦ j containing V ( M ′′ α ) whose level equals the color of M ′′ α contains V ( M ′′ α ′ ) , then either(iii) there exists an ( α − , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ ) − ψ (0 , -blocker for ( j, t ′ , − , , , t ′ ) ,or(iv) there exist t ∗ ∈ V ( T ) with i t ∗ < i t ′ and t ′ ∈ V ( T j,t ∗ ) , a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M ′′ α ) ∪ V ( M ′′ α ′ ) .Since S α − γ =1 A L ( it, − , ( V ( M ( α − γ ) ∩ X V ( T t ) − X t ⊆ ( X V ( T t ′ ) − X t ′ ) ∩ I ◦ j , M ′′ γ = M γ for every γ ∈ [ α − .For every v ∈ V ( P ) ∩ A L ( it, − , ( V ( M ( | V ( G ) | ) α )) − Y ( i t , − , , since c ( v ) = c ( P ) and S α − α =1 A L ( it, − , ( V ( M α )) ∩ X V ( T t ) − X t ⊆ I ◦ j and t κ ∈ V ( T t ′ ) − { t ′ } , we know v Y ( i t , | V ( T ) | +1 ,s +2) , so v ∈ I ◦ j and there exists q ∗ v ∈ ∂T j,t such that v ∈ X V ( T q ∗ v ) − X q ∗ v and v Y ( i q ∗ v , − , . Let v be a vertex in V ( P ) ∩ A L ( it, − , ( V ( M ( | V ( G ) | ) α )) − Y ( i t , − , such that q ∗ v = t ′ if possible.Now we prove(e) For every v ∈ V ( P ) ∩ A L ( it, − , ( V ( M ( | V ( G ) | ) α )) − Y ( i t , − , , q ∗ v = t ′ .Suppose (e) does not hold. Then q ∗ v = t ′ by the choice of v . By Claim 4.12.34,the monochromatic E ( i q ∗ v ) j,q ∗ v -pseudocomponent in G [ Y ( i q ∗ v , − , ] containing M α does not havethe smallest σ -value among all S ◦ j -related monochromatic E ( i q ∗ v ) j,q ∗ v -pseudocomponents M ′ in G [ Y ( i q ∗ v , − , ] with A L ( iq ∗ v , − , ( V ( M ′ )) ∩ X V ( T q ∗ v ) − X q ∗ v = ∅ . By Claim 4.12.31, since there ex-ists no e ∈ E ( i t , | V ( G ) | ) j,t − E ( i t ) j,t such that e ∩ S α − α =1 V ( M α ) = ∅ , there exists a c -monochromaticpath R in G + E ( i t ) j,q ∗ v from S α − α =1 V ( M α ) to X q ∗ v internally disjoint from S α − α =1 V ( M α ) .Since A L ( it, − , ( S α − α =1 V ( M α )) ∩ X V ( T t ) − X t ⊆ X V ( T tκ ) ∩ I ◦ j , there exists a subpath R ′ from V ( M ′′ α ′ ) ∩ V ( R ′ ) to V ( M ′′ α ′ ) ∩ V ( R ′ ) internally disjoint from S w α =1 V ( M ′′ α ) such that V ( R ′ ) ∩ A L ( it ′ , − , ( V ( M ′′ α ′ )) ∩ X V ( T t ′ ) − X t ′ = ∅ for some α ′ ∈ [ α − and α ′ ∈ [ α ′ + 1 , w ] .By Claim 4.12.34, α ′ ∈ [2 , α − . Let α ′′ be the minimum γ ∈ [2 , α − such that there exists α ′′ ∈ [ γ + 1 , w ] such that there exists a subpath R ′′ from V ( M ′′ γ ) ∩ V ( R ′′ ) to V ( M ′′ α ′′ ) ∩ V ( R ′′ ) internally disjoint from S w α =1 V ( M ′′ α ) such that V ( R ′′ ) ∩ A L ( it ′ , − , ( V ( M ′′ γ )) ∩ X V ( T t ′ ) − X t ′ = ∅ .Note that α ′′ α ′ < α , so (iii) and (iv) hold for every α ∈ [2 , α ′′ ] and α ′ ∈ [ α + 1 , w ] . Since σ ( M ′′ α ′′ ) σ ( M ′′ α ′ ) < σ ( M α ) , the minimality of γ implies that this claim is applicable when t is replaced by t ′ and M α is replaced by M ′′ α ′′ . Hence ( t ′ , M ′′ α ′′ ) is not an outer-safe pair.But σ ( M ′′ α ′′ ) < σ ( M α ) , so ( t ′ , M ′′ α ′′ ) is an outer-safe pair by the assumption of this claim, acontradiction. 86ence (e) holds. So M ′′ α ⊇ M ( | V ( G ) | ) α ⊇ M α and V ( P ) ∩ A L ( it ′ , − , ( V ( M ′′ α )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Note that by an argument similar with the one for showing that q ∗ v = t ′ , M ′′ α = M ( | V ( G ) | ) α .Since v Y ( i t ′ , − , , V ( P ) Y ( i t ′ , − , .Suppose there exist β (0)2 ∈ [ w ] with M (0) β M ′′ α and a subpath P ′ of P from V ( M ′′ α ) to V ( M (0) β ) internally disjoint from S w γ =1 V ( M (0) γ ) such that V ( P ′ ) ∩ A L ( it ′ , − , ( V ( M ′′ α )) ∩ X V ( T t ′ ) − X t ′ = ∅ . Note that V ( P ′ ) ⊆ X V ( T t ′ ) since X t ′ ∩ I j ⊆ Y ( i t , | V ( T ) | +1 ,s +2) . Since V ( P ′ ) isinternally disjoint from S w γ =1 V ( M (0) γ ) ⊇ S α − γ =1 V ( M ′′ γ ) , there exists α ′′ ∈ [ w ] such that M ′′ α ′′ contains M (0) β . Since P ′ is a subpath of P , α ′′ ∈ [ α , w ] . Since M β (0)2 M ′′ α , α ′′ ∈ [ α + 1 , w ] .Let α ′′ = α . For each β ∈ [2] , let Q β = { M ( α − γ : γ ∈ [ w ] , M ( α − γ ⊆ M ′′ α ′′ β } . Since α ′′ = α ,there exists no e ∈ E ( i t ,α ) j,t − E ( i t ,α − j,t such that e ∩ S Q ∈Q V ( Q ) = ∅ 6 = e ∩ S Q ∈Q V ( Q ) . Bythe construction of E ( i t ,α ) j,t , for every Q ∈ Q and Q ∈ Q , either there exists t ∗ ∈ V ( T ) with i t ∗ < i t and t ′ ∈ V ( T j,t ∗ ) , a witness q ′ ∈ ∂T j,t ∗ for X q ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M ′′ α ′ ) ∪ V ( M ′′ α ′ ) , or there exists a blocker for Q and Q . If the former holds,then by Claims 4.12.25 and 4.12.33 and the existence of P ′ , there exists a monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X q ′ and V ( M α ) , contradicting(c). So desired blockers for Q ∈ Q and Q ∈ Q exist. But (iii) and (iv) hold for every α ∈ [ α − , so Claim 4.12.40 implies that the conditions of this claim hold, when t isreplaced by t ′ and M α replaced by M ′′ α . Since σ ( M ′′ α ) σ ( M α ) and i ′ t > i t , the choice of ( t, M α ) for this claim and the existence of P ′ implies that ( t ′ , M ′′ α ) is not outer-safe. But V ( M ′′ α ) = V ( M ( | V ( G ) | ) α ) ⊆ Y ( i t , | V ( T ) | +1 ,s +2) , a contradiction.This together with (e) and the fact that M ′′ α = M ( | V ( G ) | ) α imply that M α ⊆ M ′′ α = M ( | V ( G ) | ) α . Recall that there exists α ∗ ∈ { α , α } such that A L ( it, − , ( V ( M α ∗ )) ∩ ( X V ( T t ) − X t ) ∩ X V ( T t ) − X t = ∅ . Since V ( P ) is internally disjoint from X t , by (d) and Claim 4.12.24,there exists a vertex o in V ( P ) ∩ A L ( it, | V ( T ) | +1 ,s +2) ( V ( M ( | V ( G ) | ) α )) ∩ X V ( T t ′ ) − ( X t ′ ∪ X V ( T t η ) )) .So o ∈ V ( P ) ∩ A L ( it ′ , − , ( V ( M ′′ α )) ∩ X V ( T t ′ ) − ( X t ′ ∪ X V ( T t η ) )) .Let z be the node of T with i z > i t ′ such that either o ∈ X V ( T j,z ) or there exists z ′ ∈ ∂T j,z such that o ∈ X V ( T z ′ ) − X z ′ and t κ V ( T z ′ ) , and subjec to this, i z is minimum. Since o ∈ A L ( it ′ , − , ( V ( M ′′ α )) − X V ( T t η ) , an augument similar to the one for showing (d) shows that t η V ( T j,z ) . If o ∈ X V ( T j,z ) , then since t η V ( T j,z ) and o ∈ A L ( it ′ , − , ( V ( M ′′ α )) − X V ( T t η ) , o ∈ Y ( i z , ,α ) with c ( o ) = c ( P ) , a contradiction. So there exists z ′ ∈ ∂T j,z such that o ∈ X V ( T z ′ ) − X z ′ and t κ V ( T z ′ ) . Then an argument similar to the one for showing (e) leads toa contradiction. This proves the claim. Claim 4.12.42. Let j ∈ [ |V| − . Let t ∈ V ( T ) . Let M be an S j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ ∈ V ( T ) be the node of T suchthat t ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] ,and subject to this, i t ∗ is minimum. Assume there exist t ′ ∈ ∂T j,t ∗ − V ( T t ) , u, v ∈ X t ′ , a -monochromatic path P t ′ in G [ X V ( T t ′ ) ] but not in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] from u to v internallydisjoint from X t ′ such that there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) = σ ( M ) containing u but not v . Let M ′ be the monochromatic E ( i t ′ ) j,t ′ -pseudocomponent in G [ Y ( i t ′ , − , ] containing M ∗ .Assume that ( z, M z ) is a nice pair and an outer-safe pair for every z ∈ V ( T ) and S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M z in G [ Y ( i z , − , ] intersecting X z with σ ( M z ) < σ ( M ) .If ( t ′ , M ′ ) is an outer-safe pair, then V ( P t ′ ) ⊆ Y ( i t ′ , − , .Proof. Suppose V ( P t ′ ) Y ( i t ′ , − , .Since t ′ ∈ ∂T j,t ∗ − V ( T t ) , t = t ∗ = t ′ . Since M ′ contains M ∗ , σ ( M ′ ) σ ( M ∗ ) = σ ( M ) .Since P t ′ is a c -monochromatic path and V ( P t ′ ) Y ( i t ′ , − , , by Claim 4.12.34, there existsan S ◦ j -related monochromatic E ( i t ′ ) j,t ′ -pseudocomponent Q ′ in G [ Y ( i t ′ , − , ] intersecting X t ′ with A L ( it ′ , − , ( V ( Q ′ )) ∩ X V ( T t ′ ) − X t ′ = ∅ and σ ( Q ′ ) < σ ( M ′ ) . Since σ ( Q ′ ) < σ ( M ′ ) σ ( M ∗ ) = σ ( M ) , ( t ′ , Q ′ ) is an outer-safe pair and a nice pair. By Claim 4.12.33, t ′ is a witness for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] . Let q ∗ ∈ V ( T ) be the node of T such that t ′ ∈ V ( T j,q ∗ ) and some E ( i q ∗ ) j,q ∗ -pseudocomponent in G [ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( Q ′ ) and X q for some witness q ∈ ∂T j,q ∗ ∪ { q ∗ } for X q ∩ I j ⊆ W ( i q ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subjectto this, i q ∗ is minimum. Since t ∗ is a candidate for q ∗ , i q ∗ i t ∗ . So by Claim 4.12.36, thereexists an S ◦ j -related monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent Q in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting X t ′ with A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( Q )) ∩ X V ( T t ′ ) − X t ′ = ∅ and σ ( Q ) = σ ( Q ′ ) < σ ( M ∗ ) .For every ℓ ∈ [0 , | V ( G ) | − and x ∈ { u, v } , let Q ( ℓ ) x be the monochromatic E ( i t ∗ ,ℓ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] containing x . For every ℓ ∈ [0 , | V ( G ) | − , let Q ( ℓ ) be the monochromatic E ( i t ∗ ,ℓ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] such that σ ( Q ( ℓ ) ) is the ( ℓ +1) -th smallest among all monochromatic E ( i t ∗ ,ℓ ) j,t ∗ -pseudocomponents in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] .Let L = { ℓ ∈ [0 , | V ( G ) | − 1] : Q ( ℓ ) is S ◦ j -related , V ( Q ( ℓ ) ) ∩ X t ′ = ∅ , A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( Q ( ℓ ) )) ∩ X V ( T t ′ ) − X t ′ = ∅ , σ ( Q ( ℓ ) ) < min { Q ( ℓ ) u , Q ( ℓ ) v }} . By the existenceof Q , L 6 = ∅ .For each ℓ ∈ L , let M ( ℓ )1 , M ( ℓ )2 , . . . , M ( ℓ ) β (for some β ∈ N ) be the S ◦ j -related monochromatic E ( i t ∗ ,ℓ ) j,t ∗ -pseudocomponents M ′′ intersecting X t ′ with A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′′ )) ∩ X V ( T t ′ ) − X t ′ = ∅ such that σ ( M ( ℓ )1 ) < σ ( M ( ℓ )2 ) < · · · < σ ( M ( ℓ ) β ) and V ( M ( ℓ ) γ ) = ∅ for every γ > β + 1 . Let m L = max L .Let M ′ , M ′ , . . . , M ′ β (for some β ′ ∈ N ) be the S ◦ j -related monochromatic E ( i t ′ ) j,t ′ -pseudocomponents M ′′ intersecting X t ′ with A L ( it ′ , − , ( V ( M ′′ )) ∩ X V ( T t ′ ) − X t ′ = ∅ suchthat σ ( M ′ ) < σ ( M ′ ) < · · · < σ ( M ′ β ′ ) . Let V ( M ′ γ ) = ∅ for every γ > β ′ + 1 .Let α ∈ [ w ] be the minimum index such that V ( M ′ α ) ∩ { u, v } 6 = ∅ . Since M ∗ con-tains u but not v , there exists no monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent containingboth u and v . So by the construction of E ( i t ∗ , | V ( G ) | ) j,t ∗ , ( V ( Q ( m L ) u ) ∪ V ( Q ( m L ) v )) ∩ X t ∗ = ∅ and S M ′ A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ∗ ) − X t ∗ ⊆ ( X V ( T t ′ ) − X t ′ ) ∩ I ◦ j , where the union is overall S ◦ j -related E ( i t ∗ ,m L ) j,t ∗ -pseudocomponents M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with V ( M ′ ) ∩ X t ′ = ∅ , A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ∗ ) − X t ∗ = ∅ and σ ( M ′ ) σ ( Q ( m L ) ) . So by Claim 4.12.31,we know for every α ∈ [ α − , M ′ α = M ( | V ( G ) | ) α by induction on α . This together with88laim 4.12.31 imply that M ′ α = M ( | V ( G ) | ) α . So | M ′ α ∩ { u, v }| = 1 . Hence there exists α ∈ [ α + 1 , w ] such that M ′ α ∩ { u, v } 6 = ∅ . Note that M ∗ is contained in M ′ α or M ′ α . So σ ( M ∗ ) σ ( M ′ α ) and M ′ ∈ { M ′ α , M ′ α } .For ℓ ∈ [2] , let Q α = { M ( m L ) γ : γ ∈ [ w ] , M ( m L ) γ ⊆ M ′ α ℓ } . By the existence of P t ′ , the minimality of i t ∗ and the construction of E ( i t ∗ , | V ( G ) | ) j,t ∗ , for every M ( m L ) γ ∈ Q and M ( m L ) γ ∈ Q , we have ( V ( M ( m L ) γ ) ∪ V ( M ( m L ) γ )) ∩ X t ∗ = ∅ and there exists an ( m L + 1 , γ , γ , ψ ( γ , γ ) , ψ ( γ , γ )) -blocker for ( j, t ∗ , | V ( T ) | + 1 , s + 2 , m L , t ′ ) . Hence,by Claim 4.12.31, V ( M ′ α ) ∩ X t ∗ = ∅ . Since σ ( M ′ α ) σ ( M ∗ ) = σ ( M ) , for ev-ery β ∈ [ α − , ( t ′ , M ′ β ) is a nice pair. Hence by Claim 4.12.40, there exists an ( α − , α , α , ψ ( α , α ) , ψ ( α , α ) − ψ (0 , -blocker for ( j, t ′ , − , , , t ′ ) .Similarly, by the construction of E ( i t ∗ ,m L ) j,t ∗ and Claim 4.12.40, for every α ∈ [2 , α − and α ′ ∈ [ α + 1 , w ] , if M ′ α and M ′ α ′ have the same color, and the s -segment in S ◦ j containing V ( M ′ α ) whose level equals the color of M ′ α contains V ( M ′ α ′ ) , then either there exists an ( α − , α, α ′ , ψ ( α, α ′ ) , ψ ( α, α ′ ) − ψ (0 , -blocker for ( j, t ′ , − , , , t ′ ) , or there exist t ′′′ ∈ V ( T ) with i t ′′′ < i t ∗ and T j,t ∗ ⊆ T j,t ′′′ , a witness q ′ ∈ ∂T j,t ′′′ for X q ′ ∩ I j ⊆ W ( i t ′′′ ,ℓ )3 forsome ℓ ∈ [0 , | V ( T ) | ] , and a monochromatic E ( i t ′′′ ) j,t ′′′ -pseudocomponent in G [ Y ( i t ′′′ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ′ α ) ∪ V ( M ′ α ′ ) .Since V ( P t ′ ) Y ( i t ′ , − , , by Claim 4.12.34, α = 1 . If M ′ α = M ′ , then ( t ′ , M ′ α ) = ( t ′ , M ′ ) is an outer-safe pair; if M ′ α = M ′ , then M ′ = M ′ α , so σ ( M ′ α ) < σ ( M ′ α ) = σ ( M ′ ) σ ( M ∗ ) = σ ( M ) , and hence ( t ′ , M ′ α ) is an outer-safe pair. By Claim 4.12.41, there exists no c -monochromatic path O ∗ in G from V ( M ′ α ) ∩ V ( O ∗ ) to V ( M ′ α ) ∩ V ( O ∗ ) internally disjointfrom S w γ =1 V ( M ′ γ ) such that V ( O ∗ ) ∩ A L ( it ′ , − , ( V ( M ′ α )) ∩ X V ( T t ′ ) − X t ′ = ∅ , contradictingthe existence of P t ′ . This proves the claim. Claim 4.12.43. Let j ∈ [ |V| − . Let t ∈ V ( T ) . Let M be an S j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Assume that ( z, M z ) is a nice pair andan outer-safe pair for every z ∈ V ( T ) and S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M z in G [ Y ( i z , − , ] intersecting X z with σ ( M z ) < σ ( M ) . Then ( t, M ) is an outer-safe pair.Proof. Suppose to the contrary that ( t, M ) is not an outer-safe pair, and subject to this, i t isas small as possible.Let t ∗ ∈ V ( T ) be the node of T such that t ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponentin G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ) and X q for some witness q ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i t ∗ is minimum. Since ( t, M ) is notan outer-safe pair, there exist t ′ ∈ ∂T j,t ∗ − V ( T t ) , u, v ∈ X t ′ , a monochromatic path P t ′ in G [ Y ( i t , − , ∩ X V ( T t ′ ) ] but not in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] from u to v internally disjoint from X t ′ suchthat there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] with σ ( M ∗ ) = σ ( M ) containing u but not v . In particular, t = t ∗ . By Claim 4.12.33, t ′ is awitness for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] .Let M ′ be the monochromatic E ( i t ′ ) j,t ′ -pseudocomponent in G [ Y ( i t ′ , − , ] containing M ∗ . ByClaims 4.12.25 and 4.12.33, t ∗ ∈ V ( T ) is the node of T such that t ′ ∈ V ( T j,t ∗ ) and some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersects V ( M ′ ) and X q for some witness89 ∈ ∂T j,t ∗ ∪ { t ∗ } for X q ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i t ∗ isminimum. Note that t ′ = t ∗ and σ ( M ′ ) σ ( M ∗ ) .Suppose V ( P t ′ ) ⊆ Y ( i t ′ , − , . If σ ( M ′ ) = σ ( M ∗ ) , then since t ′ ∈ ∂T j,t ∗ ∩ V ( T t ′ ) , it is acontradiction by Claim 4.12.37. So σ ( M ′ ) < σ ( M ∗ ) = σ ( M ) . Hence ( t ′ , M ′ ) is an outer-safepair by assumption. By Claim 4.12.38, there uniquely exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent Q ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ′ ) . Since M ∗ ⊆ M ′ , Q ∗ = M ∗ bythe uniqueness of Q ∗ . By Claim 4.12.38, σ ( Q ∗ ) = σ ( M ′ ) . So σ ( M ′ ) = σ ( M ∗ ) , a contradiction.Hence V ( P t ′ ) Y ( i t ′ , − , . By Claim 4.12.42, ( t ′ , M ′ ) is not an outer-safe pair. By theassumption of this claim, σ ( M ′ ) > σ ( M ) . Since M ′ ⊇ M ∗ , σ ( M ′ ) σ ( M ∗ ) = σ ( M ) . So σ ( M ′ ) = σ ( M ) . Since ( t ′ , M ′ ) is not outer-safe and σ ( M ′ ) = σ ( M ) , by the maximality of i t , i t ′ > i t . But V ( P t ′ ) ⊆ Y ( i t , − , and V ( P t ′ ) Y ( i t ′ , − , . So i t > i t ′ , a contradiction. Thisproves the claim. Claim 4.12.44. Let j ∈ [ |V| − . Let t ∈ V ( T ) . Let M be an S j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Assume that ( z, M z ) is a nice pair andan outer-safe pair for every z ∈ V ( T ) and S ◦ j -related monochromatic E ( i z ) j,z -pseudocomponent M z in G [ Y ( i z , − , ] intersecting X z with σ ( M z ) < σ ( M ) . Then ( t, M ) is a nice pair.Proof. By Claim 4.12.43, ( t, M ) is an outer-safe pair. Let t ∗ be the node of T with t ∈ V ( T j,t ∗ ) such that some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) intersects X t ′ for some witness t ′ ∈ ∂T j,t ∗ ∪ { t ∗ } for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i t ∗ is minimum. By Claim 4.12.38, there uniquely ex-ists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) ; V ( M ∗ ) ∩ X t = ∅ and σ ( M ∗ ) = σ ( M ) ; if t = t ∗ and A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ , then A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ∗ )) ∩ X V ( T t ) − X t = ∅ .Suppose to the contrary that ( t, M ) is not a nice pair. So t = t ∗ , and there exista c -monochromatic path P in G intersecting V ( M ) with V ( P ) ⊆ X V ( T t ∗ ) internally dis-joint from X V ( T t ) and a vertex u P ∈ V ( P ) ∩ X V ( T j,t ∗ ) such that there exists no monochro-matic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] such that V ( M ′ ) ∩ V ( M ) = ∅ 6 = V ( M ′ ) ∩ X V ( T t ) , M ′ contains u P , and if A L ( it, − , ( V ( M )) ∩ X V ( T t ) − X t = ∅ , then A L ( it ∗ , | V ( T ) | +1 ,s +2) ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ . So M ∗ does not contain u P . By Claims 4.12.25and 4.12.33, some monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent Q in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] con-tains u P . Hence Q = M ∗ .Since V ( P ) ∩ V ( M ) = ∅ and V ( P ) ⊆ X V ( T j,t ∗ ) , V ( P ) ∩ V ( M ∗ ) = ∅ by the uniqueness of M ∗ .Since P is internally disjoint from X V ( T t ) , by tracing along P from M ∗ , there exists t ′ ∈ ∂T j,t ∗ − V ( T t ) , u, v ∈ X t ′ , a c -monochromatic path P t ′ in G [ X V ( T t ′ ) ] but not in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] from u to v internally disjoint from X t ′ such that M ∗ contains u but not v . Since M ∗ doesnot contain v , M does not contain v by Claims 4.12.25 and 4.12.33 and the uniquenessof M ∗ . So V ( P t ′ ) Y ( i t , − , . Let M ′ be the monochromatic E ( i t ′ ) j,t ′ -pseudocomponent in G [ Y ( i t ′ , − , ] containing M ∗ . So σ ( M ′ ) σ ( M ∗ ) = σ ( M ) . Hence ( t ′ , M ′ ) is an outer-safe pairby Claim 4.12.43. By Claim 4.12.42, V ( P t ′ ) ⊆ Y ( i t ′ , − , . Hence ( t, M ) is not outer-safe, acontradiction. This proves the claim. 90 laim 4.12.45. Let j ∈ [ |V|− . Let t ∈ V ( T ) . Let M be an S ◦ j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ be the node of T with t ∈ V ( T j,t ∗ ) such that some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) intersects X t ′ for some witness t ′ ∈ ∂T j,t ∗ ∪ { t ∗ } for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , andsubject to this, i t ∗ is minimum. Then • ( t, M ) is a nice pair and an outer-safe pair, and • if P is a c -monochromatic path in G [ X V ( T t ∗ ) ] intersecting V ( M ) ∩ X t internally disjointfrom X V ( T t ) and u P is a vertex in V ( P ) ∩ X V ( T j,t ∗ ) , then there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] containing V ( M ) ∩ V ( P ) ∩ X t andcontaining u P .Proof. Suppose that ( t, M ) is either not a nice pair or not an outer-safe pair, and subjectto this, σ ( M ) is as small as possible. By Claim 4.12.43, ( t, M ) is an outer-safe pair. ByClaim 4.12.44, ( t, M ) is a nice pair. Hence the first statement of this claim holds.Now we prove the second statement of this claim. By Claim 4.12.38, there exists uniquely amonochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) .If t = t ∗ , then since V ( P ) ⊆ X V ( T t ∗ ) and P is internally disjoint from X V ( T t ) , P is a c -monochromatic path in G [ X t ] , so M ∗ containing P by Claims 4.12.25 and 4.12.33. Hence wemay assume that t = t ∗ . Since ( t, M ) is nice, the uniqueness of M ∗ implies that V ( M ∗ ) ∩ V ( M ) = ∅ 6 = V ( M ∗ ) ∩ X V ( T t ) and M ∗ contains u P . By Claims 4.12.25, 4.12.33 and 4.12.38, V ( M ∗ ) contains V ( M ) ∩ X t , so V ( M ∗ ) contains V ( M ) ∩ V ( P ) ∩ X t . This proves the claim. Claim 4.12.46. Let j ∈ [ |V|− . Let t ∈ V ( T ) . Let M be an S j -related monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] intersecting X t . Let t ∗ be the node of T with t ∈ V ( T j,t ∗ ) suchthat some E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) intersects X t ′ forsome witness t ′ ∈ ∂T j,t ∗ ∪ { t ∗ } for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject tothis, i t ∗ is minimum. Let P be a c -monochromatic path in G intersecting V ( M ) ∩ X t internallydisjoint from X V ( T t ) . Let u P be a vertex of P . If either u P ∈ X V ( T j,t ∗ ) or σ ( u P ) = σ ( P ) , then M contains u P .Proof. Suppose this claim does not hold. Among all counterexamples, we choose ( t, M, P, u P ) such that i t is as small as possible.Suppose t = t ∗ . Let p be the parent of t . If V ( P ) ∩ X p = ∅ , then p is a candidate for t ∗ ,so t = t ∗ , a contradiction. So V ( P ) ⊆ X V ( T t ) . Hence P is a c -monochromatic path in G [ X t ] .So P ⊆ M , a contradiction.So t = t ∗ . We may assume that u P is an end of P . Let v be a vertex in V ( P ) ∩ V ( M ) .We may assume that v = u P , for otherwise we are done.By Claims 4.12.38 and 4.12.45, there uniquely exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ∗ in G [ Y ( Y ( it ∗ , | V ( T ) | +1 ,s +2) ) ] intersecting V ( M ) . By Claims 4.12.25and 4.12.33, M ∗ contains v .Let C be the collection of the maximal subpaths of P contained in G [ X V ( T j,t ∗ ) ] . Denote C by { P α : α ∈ [ |C| ] } . So for each α ∈ [ |C| ] , there exist distinct vertices v α − , u α ∈ V ( P ) such that P α is from v α − to u α . By renaming, we may assume that P passes through v , u , v , u , v , . . . , u |C| , u P in the order listed. Note that for every α ∈ [ |C| − , there exist91 α ∈ ∂T j,t ∗ ∪ { t ∗ } and a subpath Q α of P from u α to v α internally disjoint from X t α such that V ( Q α ) − X V ( T j,t ∗ ) = ∅ . Since P is internally disjoint from X V ( T t ) , t α V ( T t ) .We shall prove that for every α ∈ [ |C| − , { u α , v α , u |C| } ⊆ V ( M ∗ ) by induction on α . Foreach α ∈ [ |C| ] , since Q α is in G [ X V ( T j,t ∗ ) ] , we know if M ∗ contains v α − , then by Claims 4.12.25and 4.12.33, M ∗ contains u α . In particular, u ∈ V ( M ∗ ) . So it suffices to show that for every α ∈ [ |C| − , if M ∗ contains u α , then M ∗ contains v α .Now we fix α to be an element in [ |C|− , and assume that M ∗ contains u α . We first assume t α = t ∗ . Let Z u and Z v be the monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponents in G [ Y ( i t ∗ , − , ] containing u α and v α , respectively. Let v Q α be the vertex of Q α with σ ( v Q α ) = σ ( Q α ) . Foreach x ∈ { u α , v α } , let Q x be the subpath between x and v Q α . Note that each Q x is internallydisjoint from X V ( T t ∗ ) . Since i t ∗ < i t , both Z u and Z v contain v Q α , so Z u = Z v . Since M ∗ contains u α , M ∗ contains Z u = Z v , so M ∗ contains v α .So we may assume t α ∈ ∂T j,t ∗ − V ( T t ) . Hence Q α is a c -monochromatic path in G [ X V ( T t ∗ ) ] internally disjoint from X V ( T t ) , and v α is a vertex of P in X V ( T j,t ∗ ) . And Q α contains u α ∈ V ( M ∗ ) ⊆ V ( M ) . Since ( t, M ) is a nice pair, there exists a monochromatic E ( i t ∗ , | V ( G ) | ) j,t ∗ -pseudocomponent M ′ in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] such that V ( M ′ ) ∩ V ( M ) = ∅ and M ′ contains v α . By the uniqueness of M ∗ , M ′ = M ∗ . So M ∗ contains v α .This shows that for every α ∈ [ |C| − , { u α , v α , u |C| } ⊆ V ( M ∗ ) . Note that if u P ∈ X V ( T j,t ∗ ) , then u P = u |C| ∈ V ( M ∗ ) ⊆ V ( M ) , a contradiction. So u P X V ( T j,t ∗ ) . Hence σ ( u P ) = σ ( P ) , and there exists a subpath P ∗ of P from u |C| to u P internally disjoint from X V ( T t ∗ ) . Let M be the monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , − , ] containing u |C| . So V ( P ∗ ) ∩ V ( M ) ∩ X t ∗ = ∅ and σ ( u P ) = σ ( P ∗ ) . Since i t ∗ < i t , M contains u P . Since M contains u |C| , M ⊇ M . So M contains u P . This proves the claim. Claim 4.12.47. Let j ∈ [ |V| − . Let t ∈ V ( T ) . Let M and M be S j -related monochromatic E ( i t ) j,t -pseudocomponents in G [ Y ( i t , − , ] intersecting X t . Let P be a c -monochromatic path from V ( M ) to V ( M ) internally disjoint from X V ( T t ) . Then M = M .Proof. Let v P be the vertex of P such that σ ( v P ) = σ ( P ) . By Claim 4.12.46, both M and M contain v P . So M = M . Claim 4.12.48. Let j ∈ [ |V| − . Let t , t ∈ V ( T ) with t ∈ V ( T t ) . For each α ∈ [2] ,let M α be an S j -related monochromatic E ( i tα ) j,t α -pseudocomponent in G [ Y ( i tα , − , ] intersecting X t α . Let P be a c -monochromatic path in G from V ( M ) ∩ X t to V ( M ) internally disjointfrom X V ( T t ) . Then M ⊇ M .Proof. Suppose to the contrary that M M . Among all counterexamples, we choose ( t , t , M , M , P ) such that i t is as small as possible. By Claim 4.12.47, t ∈ V ( T t ) − { t } .So by taking a subpath, we may assume that P is internally disjoint from V ( M ) . Let v be the end of P in M , and let v be the end of P in V ( M ) ∩ X t . Note that v V ( M ) ,for otherwise M ⊇ M . By Claim 4.12.47, v ∈ X V ( T t ) .Let t ∗ be the node of T with t ∈ V ( T j,t ∗ ) such that some monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , | V ( T ) | +1 ,s +2) ] intersecting V ( M ) intersects X t ′ for some witness t ′ ∈ ∂T j,t ∗ ∪ { t ∗ } for X t ′ ∩ I j ⊆ W ( i t ∗ ,ℓ )3 for some ℓ ∈ [ − , | V ( T ) | ] , and subject to this, i t ∗ isminimum. 92uppose v ∈ X V ( T t ∗ ) . Since v V ( M ) , by Claim 4.12.46, v X V ( T j,t ∗ ) . So there exists q ∈ ∂T j,t ∗ − V ( T t ) such that v ∈ X V ( T q ) − X q , and there exists a path Q in M from v to X q internally disjoint from X q . If Q is a path in G , then by applying Claim 4.12.46 tothe path P ∪ Q , we know M contains V ( Q ) ∩ X q , so M ⊇ M , a contradiction. So Q isnot a path in G . Hence there exist q ∗ ∈ V ( T ) with T j,t ∗ ⊂ T j,q ∗ , q ′ ∈ ∂T j,q ∗ ∩ X V ( T q ) − X q and e ∈ E ( Q ) ∩ E ( i q ∗ , | V ( G ) | ) j,q ∗ = ∅ such that e ⊆ X q ′ and q ′ is a witness for X q ′ ∩ I j ⊆ W ( i q ∗ ,ℓ )3 for some ℓ ∈ [0 , | V ( T ) | ] . We may assume that there exists a subpath of Q from v to anelement in e with E ( Q ) ⊆ E ( G ) . So i q ∗ < i t ∗ . Since v ∈ X V ( T q ) − X q , there exists a subpath P of P ∪ Q from a vertex u in X q to an element of e internally disjoint from X V ( T q ) with E ( P ) ⊆ E ( G ) . So V ( P ) ∩ X q ′ = ∅ . Since u ∈ X V ( T j,t ∗ ) , u ∈ V ( M ) by Claim 4.12.46. Bythe existence of P and Claims 4.12.25 and 4.12.33, V ( M ) ∩ X q ′ = ∅ , so q ∗ is a candidate for t ∗ . But i q ∗ < i t ∗ , a contradiction.Hence v X V ( T t ∗ ) . By the existence of P , the parent of t is a candidate for t ∗ , so t ∗ = t .Let P ′ be the maximal subpath of P from v to X t ∗ . Let u be the end of P ′ other than v if possible. Since u ∈ X t ∗ ⊆ X V ( T j,t ∗ ) , by Claim 4.12.46, M contains u . Let M u bethe monochromatic E ( i t ∗ ) j,t ∗ -pseudocomponent in G [ Y ( i t ∗ , − , ] containing u . So there exists asubpath P ′′ of P from u to v internally disjoint from X V ( T t ∗ ) . Since v ∈ X V ( T t ) − X V ( T t ∗ ) , t ∗ ∈ V ( T t ) . Since i t ∗ < i t , the minimality of i t implies that M u ⊇ M . Since M contains u ∈ V ( M u ) , M ⊇ M u ⊇ M , a contradiction. This proves the claim.A parade ( t , t , . . . , t k ) is tamed if for every α ∈ [ k − , t α +1 ∈ ∂T j,t α . Claim 4.12.49. Let j ∈ [ |V| − . Let κ = κ . Let ( t , t , . . . , t κ ) be a parade that is asubsequence of a tamed parade. For every β ∈ [ κ ] , let p ( β ) = ( p ( β )1 , p ( β )2 , . . . , p ( β ) | V ( G ) | ) be the ( i t β , − , , j ) -pseudosignature. Let α ∈ [ | V ( G ) | ] such that p (1) α is not a zero sequence. Let M be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining p (1) α . For every β ∈ [2 , κ ] , let M β be the monochromatic E ( i tβ ) j,t β -pseudocomponent in G [ Y ( i tβ , − , ] containing M . For every α ′ ∈ [ α − in which p (1) α ′ is a nonzero sequence, let Q (1) α ′ be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining p (1) α ′ , and for every β ∈ [2 , κ ] , let Q ( β ) α ′ bethe monochromatic E ( i tβ ) j,t β -pseudocomponent in G [ Y ( i tβ , − , ] containing Q (1) α ′ . Assume that forevery α ′ ∈ [ α − in which p (1) α ′ is a nonzero sequence, V ( Q (1) α ′ ) = V ( Q ( κ ) α ′ ) . If V ( M β +1 ) = V ( M β ) for every β ∈ [ κ − , then there exists α ∗ ∈ [ α − such that p ( κ ) α ∗ is lexicographicallysmaller than p (1) α ∗ , and p ( κ ) β = p (1) β for every β ∈ [ α ∗ − Proof. Suppose to the contrary that this claim does not hold. We may assume that amongall counterexamples, α is minimal.Since for every α ′ ∈ [ α − in which p (1) α ′ is a nonzero sequence, V ( Q (1) α ′ ) = V ( Q ( κ ) α ′ ) ,we know for every β ∈ [ κ ] , V ( Q ( β ) α ′ ) = V ( Q (1) α ′ ) . So, together with Claim 4.12.32, for every α ′ ∈ [ α − , β ∈ [ κ − and γ ∈ [ | V ( G ) | ] , the γ -th entry of p ( β +1) α ′ is at most the γ -th entryof p ( β ) α . Hence if there exist β ∈ [2 , κ ] and α ′ ∈ [ α − such that p ( β ) α ′ = p (1) α ′ , then p ( κ ) α ′ islexicographically smaller than p (1) α ′ and we are done if we further choose α ′ to be minimum.93o for every β ∈ [ κ ] and α ′ ∈ [ α − , p ( β ) α ′ = p (1) α ′ . Hence for every β ∈ [ κ ] and α ′ ∈ [ α − inwhich p (1) α ′ is a nonzero sequence, Q ( β ) α ′ defines p ( β ) α ′ , and A L ( it , − , ( V ( Q (1) α ′ )) ∩ X V ( T t ) − X t = A L ( itβ , − , ( V ( Q ( β ) α ′ )) ∩ X V ( T tβ ) − X t β ⊆ X V ( T tκ ) − X t κ . For every α ′ ∈ [ α − , since V ( Q (1) α ′ ) ∩ X t = ∅ , we know A L ( it , − , ( V ( Q (1) α ′ )) ∩ X V ( T t ) − X t = A L ( itκ , − , ( V ( Q ( κ ) α ′ )) ∩ X V ( T tκ ) − X t κ ⊆ X V ( T tκ ) − ( X t κ ∪ Z t ) ⊆ ( X V ( T tκ ) − X t κ ) ∩ I ◦ j . Since ( t , . . . , t κ ) is a subsequence of a tamedparade, for every β ∈ [ κ − and α ′ ∈ [ α − , ( A L ( itβ , − , ( V ( Q ( β ) α ′ )) ∩ X V ( T tβ ) − X t β ) ∩ Z t β = ∅ .Let κ ′ = κ + w + 3 . Since A L ( itβ , − , ( V ( Q ( β ) α ′ )) ∩ X V ( T tβ ) − X t β ⊆ ( X V ( T tκ ) − X t κ ) ∩ I ◦ j for every β ∈ [ κ − and α ′ ∈ [ α − , we know that for any distinct β , β ∈ [ κ ] with t β ∈ V ( T t β ) − { t β } , if X t β ∩ I j ⊆ X t β ∩ I j , then since M β = M β , some element in E ( M β ) − E ( M β ) is a fake edge contained in X t β ∩ I j . So for every β ∈ [ κ ] , there are at most (cid:0) w (cid:1) elements β ′ ∈ [ β + 1 , κ ] such that X t β ′ ∩ I j ⊆ X t β ∩ I j . Hence there exists a subsequence ( t j , t j , ... ) of ( t , t , ..., t κ ) of length at least κ/w such that for any distinct β , β ∈ [ ⌈ κ/w ⌉ ] with β < β , X t jβ ∩ I j X t jβ ∩ I j . Since κ/w > κ /w > N . w , κ ′ ) and ( t , t , . . . , t κ ) is a parade with | X t β ∩ I j | w for every β ∈ [ κ ] , by Lemma 4.10, there exists a ( t ′ , t ′ κ ′ , ℓ ∗ ) -fanin ( T, X | G [ I j ] ) of size κ ′ (for some ℓ ∗ ∈ [0 , w ] ) that is a subsequence of ( t , t , . . . , t κ ) suchthat X t ′ β ∩ I j − X t ′ = ∅ for every β ∈ [ κ ′ ] .Since ( t ′ , t ′ , . . . , t ′ κ ′ ) is a subsequence of ( t , t , . . . , t κ ) , for simplicity, by removing somenodes in ( t , t , . . . , t κ ) , we may assume ( t , t , . . . , t κ ′ ) = ( t ′ , t ′ , . . . , t ′ κ ′ ) .Let β ∗ ∈ [ κ ′ − κ − . Since V ( M β ∗ +1 ) = V ( M β ∗ ) and A L ( itβ ∗ , − , ( V ( Q ( β ∗ ) α ′ )) ∩ X V ( T tβ ∗ ) − X t β ∗ = A L ( itβ ∗ +1 , − , ( V ( Q ( β ∗ +1) α ′ )) ∩ X V ( T tβ ∗ +1 ) − X t β ∗ +1 ⊆ ( X V ( T tκ ) − X t κ ) ∩ I ◦ j for every α ′ ∈ [ α − , there exists e β ∗ ∈ E ( M β ∗ +1 ) ∩ E ( i tβ ∗ +1 ) j,t β ∗ +1 − E ( M β ∗ ) such that | e ∩ V ( M β ∗ ) | = 1 . Sothere exists z ∈ V ( T ) with T t β ∗ +1 ⊆ T z ⊆ T t β ∗ and i t β ∗ i z < i t β ∗ +1 such that e β ∗ ∈ E ( i z , | V ( G ) | ) j,z − E ( i z ) j,z . Hence there exists ℓ ∈ [ | V ( G ) | ] such that e β ∗ ∈ E ( i z ,ℓ ) j,z − E ( i z ,ℓ − j,z . Let t ′ ∈ ∂T j,z be the witness for X t ′ ∩ I j ⊆ W ( i z ,ℓ ′ )3 for some ℓ ′ ∈ [0 , | V ( T ) | ] such that e β ∗ ⊆ X t ′ .Since V ( M β ∗ ) ∩ X t = ∅ 6 = V ( M β ∗ ) ∩ X t ′ , V ( M β ∗ ) ∩ X z = ∅ .Denote e β ∗ by { u, v } . Let M u , M v be the monochromatic E ( i z ,ℓ ) j,z -pseudocomponents in G [ Y ( i z , | V ( T ) | +1 ,s +2) ] containing u, v , respectively. Let M be the S ◦ j -related monochromatic E ( i z ,ℓ ) j,z -pseudocomponent in G [ Y ( i z , | V ( T ) | +1 ,s +2) ] such that σ ( M ) is the ( ℓ + 1) -th smallestamong all monochromatic E ( i z ,ℓ ) j,z -pseudocomponents in G [ Y ( i z , | V ( T ) | +1 ,s +2) ] . So σ ( M ) < min { σ ( M u ) , σ ( M v ) } and M is S ◦ j -related. Since M β ∗ contains u or v , M β ∗ is contained inone of M u and M v . So σ ( M ) < σ ( M β ∗ ) .Since β ∗ κ ′ − κ − and ( t , . . . , t κ ′ ) is a subsequence of a tamed parade, t κ ′ − κ V ( T j,z ) .For every α ′ ∈ [ α − in which p (1) α ′ is a nonzero sequence, let Q ( z ) α ′ be the monochromatic E ( i z ) j,z -pseudocomponent in G [ Y ( i z , − , ] containing Q (1) α ′ . Since i z < i t κ ′− κ − , for every β ∈ [ κ ′ − κ − and α ′ ∈ [ α − , A L ( iz, | V ( T ) | +1 ,s +2) ( V ( Q ( z ) α ′ )) ∩ X V ( T z ) − X z ⊆ X V ( T tκ ′ ) − ( X t κ ′ ∪ Z z ) . ByClaim 4.12.31, since Q (1) α ′ = Q ( z ) α ′ = Q ( κ ) α ′ for every α ′ ∈ [ α − , we know t κ ′ − κ ∈ V ( T t ′ ) − { t ′ } and { Q ( z ) α ′ : α ′ ∈ [ α − , p (1) α ′ is a nonzero sequence } contains the set of the monochromatic E ( i z ,ℓ ) j,z -pseudocomponents M ′ in G [ Y ( i z , | V ( T ) | +1 ,s +2) ] with σ ( M ′ ) σ ( M ) . So E ( i z ,ℓ ) j,z − E ( i z ) j,z ⊆ X t ′ .Since ( t , t , . . . , t κ ′ ) is a ( t , t κ ′ , ℓ ∗ ) -fan in ( T, X | G [ I j ] ) , for every κ ′ − κ β < β κ ′ ,94 t β ∩ I j − X t ′ and X t β ∩ I j − X t ′ are disjoint non-empty sets. Since ( V ( M u ) ∪ V ( M v )) ∩ X z ⊇ V ( M β ∗ ) ∩ X z = ∅ and e β ∗ ∈ E ( i z ,ℓ ) j,z − E ( i z ,ℓ − j,z , we know that for every x ∈ { u, v } , A L ( iz, | V ( T ) | +1 ,s +2) ( V ( M x )) ∩ ( X V ( T z ) − X z ) ∩ X V ( T tκ ′− ( ψ w ,w ψ w ,w ) − X t κ ′− ( ψ w ,w ψ w ,w = A L ( iz, | V ( T ) | +1 ,s +2) ( V ( M x )) ∩ ( X V ( T z ) − X z ) ∩ X V ( T tκ ′− κ ) − X t κ ′− κ = ∅ , so V ( M x ) ∩ X t κ ′− κ ∩ I j = ∅ . Since | e β ∗ ∩ V ( M β ∗ ) | = 1 , ( V ( M β ∗ +1 ) − V ( M β ∗ )) ∩ X t κ ′− κ ∩ I j = ∅ .Since β ∗ is an arbitrary element in [ κ ′ − κ − , we know | X t κ ∩ I j | > κ ′ − κ − > w + 1 ,a contradiction. This proves the claim. Claim 4.12.50. Let j ∈ [ |V|− . Let γ ∗ ∈ [ w ] and κ ∈ N . Let ( t , t , . . . , t κ ) be a parade thatis a subsequence of a tamed parade. For every β ∈ [ κ ] , let p ( β ) = ( p ( β )1 , p ( β )2 , . . . , p ( β ) | V ( G ) | ) be the ( i t β , − , , j ) -pseudosignature. Let α ∈ [ | V ( G ) | ] such that p (1) α is not a zero sequence, and α isthe γ ∗ -th smallest index ℓ such that p (1) ℓ is a nonzero sequence. Let M be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining p (1) α . For every β ∈ [2 , κ ] , let M β be themonochromatic E ( i tβ ) j,t β -pseudocomponent in G [ Y ( i tβ , − , ] containing M . For every α ′ ∈ [ α − in which p (1) α ′ is a nonzero sequence, let Q (1) α ′ be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining p (1) α ′ , and for every β ∈ [2 , κ ] , let Q ( β ) α ′ be the monochromatic E ( i tβ ) j,t β -pseudocomponent in G [ Y ( i tβ , − , ] containing Q (1) α ′ .Assume ( p ( β )1 , . . . , p ( β ) α − ) is identical for every β ∈ [ κ ] . If V ( M β +1 ) = V ( M β ) for every β ∈ [ κ ] , then κ φ ( γ ∗ ) .Proof. We shall prove this claim by induction on γ ∗ . When γ ∗ = 1 , since φ (1) = κ , thisclaim follows from Claim 4.12.49. So we may assume γ ∗ > and this claim holds for smaller γ ∗ . Suppose to the contrary that κ > φ ( γ ∗ ) . Let Z = { z , z , . . . , z γ ∗ } be the set of elementsin [ α ] such that for each ℓ ∈ [ γ ∗ ] , z ℓ is the ℓ -th smallest index ℓ ′ such that p (1) ℓ ′ is a nonzerosequence. For every α ′ ∈ [ γ ∗ ] , let S α ′ = { i ∈ [ κ − 1] : V ( Q ( β +1) z α ′ ) = V ( Q ( β ) z α ′ ) } .Suppose that for every α ′ ∈ [ γ ∗ − , | S α ′ | φ ( α ′ ) . Then there exists a set R ⊆ [ κ − of consecutive positive integers with | R | > κ − P γ ∗− α ′ =1 | S α ′ | > κ − P γ ∗− α ′ =1 φ ( α ′ ) > κ such that R ∩ S γ ∗ − α ′ =1 S α ′ = ∅ . But since | R | > κ and V ( M β +1 ) = V ( M β ) for every β ∈ R , Claim 4.12.49implies that there exists α ∗ ∈ [ α − such that p ( a ) α ∗ = p ( b ) α ∗ for some distinct a, b ∈ R ⊆ [ κ ] , acontradiction.So there exists the smallest element ξ in [ γ ∗ − such that | S ξ | > φ ( ξ ) + 1 . Sincefor every α ′ ∈ [ ξ − , | S α ′ | φ ( α ′ ) , there exists a set R ∗ ⊆ [ κ − with | R ∗ ∩ S ξ | > | S ξ | / (1 + P ξ − α ′ =1 φ ( α ′ )) > κ such that R ∗ ∩ S ξ − α ′ =1 S α ′ = ∅ and there exist no β < β < β with β , β ∈ R ∗ and β ∈ S ξ − α ′ =1 S α ′ . But since | R ∗ | > κ and V ( M β +1 ) = V ( M β ) for every β ∈ R ∗ , Claim 4.12.49 implies that there exists α ∗ ∈ [ α − such that p ( a ) α ∗ = p ( b ) α ∗ for somedistinct a, b ∈ R ∗ , a contradiction. This proves the claim.95 laim 4.12.51. Let j ∈ [ |V| − . Let γ ∗ ∈ [ w ] and κ ∈ N . Let ( t , t , . . . , t κ ) be a paradethat is a subsequence of a tamed parade. For every β ∈ [ κ ] , let p ( β ) = ( p ( β )1 , p ( β )2 , . . . , p ( β ) | V ( G ) | ) be the ( i t β , − , , j ) -pseudosignature. Let α ∈ [ | V ( G ) | ] such that p (1) α is not a zero sequence,and α is the γ ∗ -th smallest index ℓ such that p (1) ℓ is a nonzero sequence. For every α ′ ∈ [ α ] in which p (1) α ′ is a nonzero sequence, let Q (1) α ′ be the monochromatic E ( i t ) j,t -pseudocomponentin G [ Y ( i t , − , ] defining p (1) α ′ , and for every β ∈ [2 , κ ] , let Q ( β ) α ′ be the monochromatic E ( i tβ ) j,t β -pseudocomponent in G [ Y ( i tβ , − , ] containing Q (1) α ′ . If for every β ∈ [ κ ] , there exists α β ∈ [ α ] such that V ( Q ( β +1) α β ) = V ( Q ( β ) α β ) , then κ φ ( γ ∗ ) .Proof. Suppose to the contrary that κ > φ ( γ ∗ ) + 1 . Let Z = { z , z , . . . , z γ ∗ } be the subsetof [ α ] such that for each α ′ ∈ [ γ ∗ ] , z α ′ is the α ′ -th smallest index ℓ such that p (1) ℓ is a nonzerosequence. For every α ′ ∈ [ γ ∗ ] , let S α ′ = { i ∈ [2 , κ ] : p ( i ) z α ′ = p ( i − z α ′ } .Suppose there exists ξ ∈ [ γ ∗ ] such that | S ξ | > φ ( ξ ) + 1 . We choose ξ to be as smallas possible. So there exists a subset R of S ξ with | R | > | S ξ | P ξ − α ′ =1 | S α ′ | > | S ξ | P ξ − α ′ =1 φ ( α ′ ) > ( φ ( ξ ) + 1) · w f ( η ) + 1 such that R ∩ S ξ − α ′ =1 S α ′ = ∅ and there exist no distinct elements a, b ∈ R such that [ a, b ] ∩ S ξ − α ′ =1 S α ′ = ∅ . Let S ′ ξ = { β ∈ R : V ( Q ( β ) z ξ ) = V ( Q ( β − z ξ ) } . Since R ∩ S ξ − α ′ =1 S α ′ = ∅ and there exist no distinct elements a, b ∈ R such that [ a, b ] ∩ S ξ − α ′ =1 S α ′ = ∅ , | S ′ ξ | φ ( ξ ) by Claim 4.12.50. So there exists a subset R ′ of R with | R ′ | > | R || S ′ ξ | +1 > w f ( η )+1 such that R ′ ∩ S ′ ξ = ∅ and for any distinct elements a < b of R ′ , [ a, b ] ∩ S ′ ξ = ∅ . Since forevery β ∈ R ′ , β ∈ R − S ′ ξ , so p ( β ) z ξ = p ( β − z ξ and V ( Q ( β ) z ξ ) = V ( Q ( β − z ξ ) , and hence p ( β ) z ξ < p ( β − z ξ .So | R ′ | w f ( η ) by Claim 4.12.30, a contradiction.Therefore, for every α ′ ∈ [ γ ∗ ] , | S α ′ | φ ( α ′ ) . Hence there exists a subset R ′′ of [2 , κ ] with | R ′′ | > κ − P γ ∗ α ′ =1 | S α ′ | > κ − P γ ∗ α ′ =1 φ ( α ′ ) > φ ( ξ )1+ P γ ∗ α ′ =1 φ ( α ′ ) such that R ∗ consists of consecutiveintegers and R ∗ ∩ S γ ∗ α ′ =1 S α ′ = ∅ . So there exists R ∗ ⊆ R ′′ with | R ∗ | > | R ′′ | /γ ∗ > φ ( γ ∗ ) + 1 such that there exists x ∗ such that for every x ∈ R ∗ , V ( Q ( x +1) x ∗ ) = V ( Q ( x ) x ∗ ) . But it contradictsClaim 4.12.50. This proves the claim. Claim 4.12.52. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∈ S ◦ j .Let t ∈ K ∗ ( M ) such that K ∗ ( M ) ∩ V ( T t ) − { t } 6 = ∅ . Then: • There exists a monochromatic E ( i t ) j,t -pseudocomponent M ′ in G [ Y ( i t , − , ] such that V ( M ′ ) ∩ X t = ∅ 6 = V ( M ′ ) ∩ V ( M ) and A L ( it, − , ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ . • There exists a monochromatic E ( i t ) j,t -pseudocomponent M ′ in G [ Y ( i t , − , ] such that V ( M ′ ) ∩ X t = ∅ and M ′ contains v M , where v M is the vertex of M with σ ( v M ) = σ ( M ) . • For every monochromatic E ( i t ) j,t -pseudocomponent M ′ in G [ Y ( i t , − , ] with V ( M ′ ) ∩ X t = ∅ 6 = V ( M ′ ) ∩ V ( M ) , A L ( it, − , ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ .Proof. Since K ∗ ( M ) ∩ V ( T t ) − { t } 6 = ∅ , there exists t ′ ∈ K ∗ ( M ) ∩ X V ( T t ) − { t } . So there exists v ∈ V ( M ) ∩ Y ( i t ′ , | V ( T ) | +1 ,s +2) ∩ X V ( T t ′ ) − Y ( i t , | V ( T ) | +1 ,s +2) . Hence v Y ( i t , − , . Let P be a pathin M from v to v M , where v M is the vertex of M with σ ( v M ) = σ ( M ) . Since t ∈ K ∗ ( M ) , t ∈ V ( T r M ) . So V ( P ) ∩ X t = ∅ and there exists a subpath P ′ of P from v to X t internally disjointfrom X t . Since v Y ( i t , − , , there exists a monochromatic E ( i t ) j,t -pseudocomponent M ′ in96 [ Y ( i t , − , ] such that V ( M ′ ) ∩ X t = ∅ 6 = V ( M ′ ) ∩ V ( P ) and A L ( it, − , ( V ( M ′ )) ∩ X V ( T t ) − X t = ∅ .Since P ⊆ M , V ( M ′ ) ∩ V ( M ) = ∅ . This proves the first statement of this claim. In addition,since V ( P ) ∩ X t = ∅ , there exists a subpath P ′′ from v M to a vertex u ′′ in X t internallydisjoint from X V ( T t ) . Let M ′′ be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] containing u ′′ . So P ′′ is a c -monochromatic path in G intersecting V ( M ′′ ) ∩ X t internallydisjoint from X V ( T t ) . Since σ ( v M ) = σ ( M ) and P ⊆ M , σ ( v M ) = σ ( P ′′ ) . By Claim 4.12.46, M ′′ contains v M . This proves the second statement of this claim.Now we prove the third statement of this claim. Let Z = S M ′′ V ( M ′′ ) , where the unionis over all monochromatic E ( i t ) j,t -pseudocomponents in G [ Y ( i t , − , ] such that V ( M ′′ ) ∩ X t = ∅ 6 = V ( M ′′ ) ∩ V ( M ) and A L ( it, − , ( V ( M ′′ )) ∩ X V ( T t ) − X t = ∅ . By the first statment ofthis claim, Z = ∅ . Let Z = S M ′′ V ( M ′′ ) , where the union is over all monochromatic E ( i t ) j,t -pseudocomponents in G [ Y ( i t , − , ] such that V ( M ′′ ) ∩ X t = ∅ 6 = V ( M ′′ ) ∩ V ( M ) and A L ( it, − , ( V ( M ′′ )) ∩ X V ( T t ) − X t = ∅ . Suppose Z = ∅ . There exists a path P ′ in M from Z to Z internally disjoint from Z ∪ Z . Note that the neighbor u of the vertex in V ( P ′ ) ∩ Z in P ′ belongs to A L ( it, − , ( Z ) . So u X V ( T t ) − X t . Hence there exists a subpath of P ′ from Z to Z internally disjoint from Z ∪ Z ∪ X V ( T t ) . Let Q be the monochromatic E ( i t ) j,t -pseudocomponentsin G [ Y ( i t , − , ] such that V ( Q ) ⊆ Z and Q contains the vertex in Z ∩ V ( P ′ ) . Since P ′ is internally disjoint from X V ( T t ) , by Claim 4.12.47, Q equals the monochromatic E ( i t ) j,t -pseudocomponents in G [ Y ( i t , − , ] containing V ( P ) ∩ Z . So V ( Q ) ⊆ Z ∩ Z , a contradiction.Hence Z = ∅ . This proves the third statement of this claim.For any i ∈ [0 , | V ( T ) |− , j ∈ [ |V|− and c -monochromatic component M with S M ∈ S ◦ j ,the ( i, j, M ) -truncation is the sequence ( b , . . . , b | V ( G ) | ) defined as follows. • Let ( p , p , . . . , p | V ( G ) | ) be the ( i, − , , j ) -pseudosignature. • Let α ∗ be the largest index γ such that p γ is a nonzero sequence and the monochromatic E ( i t ) j,t -pseudocomponent (where t is the node of T with i t = i ) in G [ Y ( i, − , ] defining p γ intersects M . (If no such index γ exists, then define α ∗ = 0 .) • For every α ∈ [ α ∗ ] , define b α = p α . • For every α ∈ [ | V ( G ) | ] − [ α ∗ ] , define b α = 0 . Claim 4.12.53. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∈ S ◦ j .Let ( t , t , . . . , t η +3 ) be a parade such that t α ∈ K ∗ ( M ) for every α ∈ [ η + 3] . Then thereexists α ∗ ∈ [2 , η + 2] such that either • the ( i t α ∗ , j, M ) -truncation is different from the ( i t , j, M ) -truncation, or • for each α ∈ { , α ∗ } , there exists a monochromatic E ( i tα ) j,t α -pseudocomponent M α in G [ Y ( i tα , − , ] with V ( M α ) ∩ X t α = ∅ 6 = V ( M α ) ∩ V ( M ) and A L ( itα , − , ( V ( M α )) ∩ X V ( T tα ) − X t α = ∅ such that M ⊆ M α ∗ and V ( M ) = V ( M α ∗ ) .Proof. For every α ∈ [ η +2] , let C α = { V ( Q ) : Q is a monochromatic E ( i tα ) j,t α -pseudocomponent M ′ in G [ Y ( i tα , − , ] with V ( M ′ ) ∩ X t α = ∅ 6 = V ( M ′ ) ∩ V ( M ) and A L ( itα, − , ( V ( M ′ )) ∩ X V ( T tα ) − X t α = ∅} .Suppose to the contrary that this claim does not hold. So for every α ∈ [ η + 2] , the ( i t α , j, M ) -truncation equals the ( i t , j, M ) -truncation. Hence, for every α ∈ [ η + 2] and every97 α ∈ C α , there exists Q ′ ∈ C such that σ ( Q α ) = σ ( Q ′ ) , so Q α ⊇ Q ′ . Therefore for every α ∈ [ η + 2] , C α = C , for otherwise second outcome of this claim holds.For every α ∈ [2 , η + 2] , since t α ∈ K ∗ ( M ) , there exists v α ∈ V ( M ) ∩ X V ( T tα ) ∩ Y ( i tα , | V ( T ) | +1 ,s +2) − Y ( i tα − , | V ( T ) | +1 ,s +2) . So { v α : α ∈ [2 , η + 2] } is a set of η + 1 differentvertices. By Claim 4.12.24, there exists α ∗ ∈ [2 , η + 2] such that v α ∗ X V ( T tη ) .Let v M be the vertex of M such that σ ( v M ) = σ ( M ) . Let P be a path in M from v α ∗ to v M . Since t α ∗ − ∈ K ∗ ( M ) , V ( P ) ∩ X t α ∗− = ∅ . Let P ′ be the subpath of P from v α ∗ to X t α ∗− internally disjoint from X t α ∗− .Let Q be the monochromatic E ( i tα ∗− ) j,t α ∗− -pseudocomponent in G [ Y ( i tα ∗− , − , ] containing thevertex in V ( P ′ ) ∩ X t α ∗− . Since v α ∗ Y ( i tα ∗− , − , , A L ( itα ∗− , − , ( V ( Q )) ∩ V ( P ′ ) − X t α ∗− = ∅ . Since P ′ is internally disjoint from X t α ∗− , V ( Q ) ∈ C α ∗ − . Since C η +2 = C = C α ∗ − , V ( Q ) ∈ C η +2 . Since both the ( i t η , j, M ) -truncation and ( i t α ∗− , j, M ) -truncation equal the ( i t , j, M ) -truncation, A L ( itα ∗− , − , ( V ( Q )) ∩ X V ( T tα ∗− ) − X t α ∗− ⊆ X V ( T tη ) − X t η . Hence V ( P ′ ) ∩ X V ( T tη ) − X t η = ∅ . Since v α ∗ X V ( T tη ) , there exists a subpath P ′′ of P ′ from v α ∗ to X t η internally disjoint from X t η .Let Q ′ be the monochromatic E ( i tη ) j,t η -pseudocomponent in G [ Y ( i tη , − , ] contain-ing X t η ∩ V ( P ′′ ) . Since t η +3 ∈ K ∗ ( M ) ∩ X V ( T tη ) − { t η +2 } , by Claim 4.12.52, A L ( itη , − , ( V ( Q ′ )) ∩ X V ( T tη ) − X t η = ∅ . So V ( Q ′ ) ∈ C η +2 = C α ∗ − . Since v t α ∗ Y ( i tα ∗− , − , and V ( Q ′ ) ∈ C α ∗ − , v t α ∗ V ( Q ′ ) . So A L ( itα ∗− , − , ( V ( Q ′ )) ∩ V ( P ′′ ) = ∅ .Let u ∈ A L ( itα ∗− , − , ( V ( Q ′ )) ∩ V ( P ′′ ) . Since u Y ( i tα ∗− , − , , u X t α ∗− . Since boththe ( i t η , j, M ) -truncation and ( i t α ∗− , − , -truncation equal the ( i t , j, M ) -truncation, A L ( itα ∗− , − , ( V ( Q ′ )) ∩ X V ( T tα ∗− ) − X t α ∗− ⊆ X V ( T tη ) − X t η . Since V ( P ′′ ) ⊆ X V ( T tα ∗− ) , u ∈ A L ( itα ∗− , − , ( V ( Q ′ )) ∩ X V ( T tα ∗− ) − X t α ∗− ⊆ X V ( T tη ) − X t η . But P ′′ is internallydisjoint from X V ( T tη ) , a contradiction. This proves the claim. Claim 4.12.54. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∈ S ◦ j .Let t , t ∈ K ∗ ( M ) such that t ∈ V ( T t ) − { t } and K ∗ ( M ) ∩ X V ( T t ) − X t = ∅ . Let α ∗ be thelargest index ℓ such that the ℓ -th entry of the ( i t , j, M ) -truncation is not a zero sequence. For α ∈ [2] , let ( q ( α )1 , . . . , q ( α ) | V ( G ) | ) be the ( i t α , j, M ) -truncation. Let α ∈ [0 , α ∗ − . If q (1) β = q (2) β forevery β ∈ [ α ] , then either • there exists β ∗ ∈ [ α + 1 , α ∗ ] such that q (2) β ∗ is not a zero sequence, or • there exists β ∗ ∈ [ α ] such that q (1) β ∗ and q (2) β ∗ are nonzero sequences and the vertex-set of themonochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining q (2) β ∗ is different from thevertex-set of the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining q (1) β ∗ .Proof. Suppose this claim does not hold. So for every β ∈ [ α + 1 , α ∗ ] , q (2) β is a zero sequence.And for every β ∗ ∈ [ α ] , if q (1) β ∗ and q (2) β ∗ are nonzero sequences, then the vertex-set of themonochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining q (2) β ∗ equals the vertex-set ofthe monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining q (1) β ∗ .98et v M be the vertex of M such that σ ( v M ) = σ ( M ) . For every k ∈ [2] , by Claim 4.12.52,there exists a monochromatic E ( i tk ) j,t k -pseudocomponent M k in G [ Y ( i tk , − , ] with V ( M k ) ∩ X t k = ∅ containing v M . By Claim 4.12.52, for each k ∈ [2] , A L ( itk , − , ( V ( M k )) ∩ X V ( T tk ) − X t k = ∅ ,so there exists γ ∗ k ∈ [ σ ( v M )] such that q ( k ) γ ∗ k is not a zero sequence and M k defines q ( k ) γ ∗ k .By the definition of α ∗ , γ ∗ ∈ [ α ∗ ] . Since v M ∈ V ( M ) ∩ V ( M ) , M ⊆ M . So γ ∗ γ ∗ .Since q (2) γ ∗ is not a zero sequence, γ ∗ [ α + 1 , α ∗ ] . Since γ ∗ γ ∗ α ∗ , γ ∗ ∈ [ α ] . In particular, α > . Since v M ∈ V ( M ) and V ( M ) ∩ X t = ∅ , V ( M ) ∩ X t = ∅ . So by Claim 4.12.31, q (1) γ ∗ is not a zero sequence.For every γ ∈ [ | V ( G ) | ] , if q (1) γ is not a zero sequence, then let Q γ be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining q (1) γ ; otherwise, let V ( Q γ ) = ∅ .Let Z = S αγ =1 V ( Q γ ) . Since q (1) γ ∗ is not a zero sequence and γ ∗ α , Z = ∅ . Since γ ∗ ∈ [ α ] , V ( Q γ ∗ ) = V ( M ) . Since M contains v M , Q γ ∗ contains v M . Hence Z ∩ V ( M ) = ∅ .Let Z = S α ∗ γ = α +1 ( V ( Q γ )) . Since ( q (1)1 , . . . , q (1) | V ( G ) | ) is the ( i t , j, M ) -truncation, V ( Q α ∗ ) ∩ V ( M ) = ∅ . So Z ∩ V ( M ) = ∅ .Since Z ∩ V ( M ) = ∅ 6 = Z ∩ V ( M ) , there exists a path P in M from Z to Z internallydisjoint from Z ∪ Z . By Claims 4.12.47 and 4.12.52, P is contained in G [ X V ( T t ) ] andis internally disjoint from X t . Let β ∗ be the index in [ α ] such that Q β ∗ contains V ( P ) ∩ Z . Since β ∗ α , q (2) β ∗ = q (1) β ∗ is not a zero sequence. So there exists a monochromatic E ( i t ) j,t -pseudocomponent Q ∗ in G [ Y ( i t , − , ] defining q (2) β ∗ . Since β ∗ ∈ [ α ] , V ( Q ∗ ) = V ( Q β ∗ ) .Since q (1) β ∗ = q (2) β ∗ , A L ( it , − , ( V ( Q β ∗ )) ∩ X V ( T t ) − X t = A L ( it , − , ( V ( Q ∗ )) ∩ X V ( T t ) − X t ⊆ X V ( T t ) − X t . Since P is contained in G [ X V ( T t ) ] and is internally disjoint from X t , V ( P ) ∩ A L ( it , − , ( V ( Q β ∗ )) ∩ X V ( T t ) − X t = ∅ . So V ( P ) ∩ X V ( T t ) − X t = ∅ .Let v be the vertex in Z ∩ V ( P ) . Let β v be the index in [ | V ( G ) | ] such that Q β v contains v ∈ V ( P ) ⊆ V ( M ) . So β v ∈ [ α + 1 , α ∗ ] .Suppose there exists β ′′ ∈ [ | V ( G ) | ] such that q (2) β ′′ is not a zero sequence and the monochro-matic E ( i t ) j,t -pseudocomponent Q ′ in G [ Y ( i t , − , ] defining q (2) β ′′ contains Q β v . Since Q ′ con-tains Q β v , β ′′ ∈ [ β v ] . Since β ′′ β v α ∗ and q (2) β ′′ is a nonzero sequence, β ′′ ∈ [ α ] , so V ( Q ′ ) = V ( Q β ′′ ) . Hence V ( Q β ′′ ) contains V ( Q β v ) . So Z ∩ Z ⊇ V ( Q β ′′ ) ∩ V ( Q β v ) = ∅ , acontradiction.Hence there exists no β ′′ ∈ [ | V ( G ) | ] such that q (2) β ′′ is not a zero sequence and the monochro-matic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining q (2) β ′′ contains Q β v .Suppose v ∈ X V ( T t ) . Since V ( Q β v ) ∩ X t = ∅ and Q β v contains v , V ( Q β v ) ∩ X t = ∅ . So byClaim 4.12.52, there exists β ′′ ∈ [ β v ] such that q (2) β ′′ is not a zero sequence and the monochro-matic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining q (2) β ′′ contains Q β v , a contradiction.Hence v ∈ V ( G ) − X V ( T t ) . Since V ( P ) ∩ X V ( T t ) − X t = ∅ , there exists v ′ ∈ V ( P ) ∩ X t suchthat the subpath P ′ of P between v and v ′ is internally disjoint from X V ( T t ) . So there exists amonochromatic E ( i t ) j,t -pseudocomponent R in G [ Y ( i t , − , ] containing v ′ . By Claim 4.12.52,there exists γ R ∈ [ | V ( G ) | ] such that q (2) γ R is not a zero sequence, and R defines q (2) γ R . But byClaim 4.12.48, R ⊇ Q β v , a contradiction. This proves the claim.We say a sequence ( a , a , . . . , a m ) is a substring of another sequence ( b , b , . . . , b n ) (for99ome m, n ∈ N ) if there exists α ∈ [0 , n − m ] such that a i = b α + i for every i ∈ [ m ] . Claim 4.12.55. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∈ S ◦ j .Let κ ∈ N . Let ( t , t , . . . , t κ ) be a parade in T such that t α ∈ K ∗ ( M ) for every α ∈ [ κ ] . Foreach α ∈ [ κ ] , let ( a ( α )1 , . . . , a ( α ) | V ( G ) | ) be the ( i t α , j, M ) -truncation. Let α ∗ be an index such that • a (1) α ∗ is a nonzero sequence, and • for every α ∈ [ α ∗ − , a (1) α = a (2) α = · · · = a ( κ ) α .Let n be the number of indices γ ∈ [ α ∗ ] such that a (1) γ is a nonzero sequence. Then κ h ( w − n ) .Proof. Note that w − n > by Claim 4.12.30. Suppose to the contrary that this claim doesnot hold. That is, κ > h ( w − n ) + 1 . We further assume that w − n is as small as possibleamong all counterexamples.For every t ∈ { t , t , . . . , t κ } and α ∈ [ | V ( G ) | ] , let a ( t ) α = a ( β ) α , where β is the integer suchthat t = t β . For every α ∈ [ | V ( G ) | ] and t ∈ { t , t , . . . , t κ } , if a ( t ) α is not a zero sequence, thenlet M ( t ) α be the monochromatic E ( i t ) j,t -pseudocomponent in G [ Y ( i t , − , ] defining a ( t ) α ; otherwise,let V ( M ( t ) α ) = ∅ . Since each t α is in K ∗ ( M ) , every subsequence of ( t , t , . . . , t κ ) is a subse-quence of a tamed parade. For every t ∈ { t , t , . . . , t κ } , let α ∗ t be the largest index γ suchthat a ( t ) γ is a nonzero sequence.Let τ = ( κ − / ( η + 3) . By Claim 4.12.53, there exists a subsequence ( t ′ , t ′ , . . . , t ′ τ ) of ( t , t , . . . , t κ − ) such that for every α ∈ [ τ − ,(i) either the ( i t ′ α , j, M ) -truncation is different from the ( i t ′ α +1 , j, M ) -truncation, or thereexists γ α ∈ [ | V ( G ) | ] such that V ( M ( t ′ α +1 ) γ α ) = V ( M ( t ′ α ) γ α ) .Since ( t ′ , t ′ , . . . , t ′ τ ) satisfies (i), by Claim 4.12.51 and the assumption of this claim, thereexists β ∈ [ φ ( w ) + 1] such that a ∗ t ′ β > α ∗ . For each γ ∈ [ τ − φ ( w )] , we redefine t ′ γ to be t ′ β + γ − . So a ∗ t ′ > α ∗ .Let τ = ( τ − φ ( w )) / w . By Claim 4.12.32, there exists a substring ( t ′′ , t ′′ , . . . , t ′′ τ ) of ( t ′ , t ′ , . . . , t ′ τ − φ ( w ) ) such that for each α ∈ [ α ∗ t ′ ] ,(ii) either a ( t ) α is a zero sequence for every t ∈ { t ′′ , t ′′ , . . . , t ′′ τ } , or a ( t ) α is a nonzero sequencefor every t ∈ { t ′′ , t ′′ , . . . , t ′′ τ } .Since ( t ′′ , t ′′ , . . . , t ′′ τ ) is a substring of ( t ′ , t ′ , . . . , t ′ τ ) , ( t ′′ , t ′′ , . . . , t ′′ τ ) satisfies (i).Let τ = τ / ( φ ( w ) + 1) . By Claim 4.12.51, there exists a substring ( t ′′′ , t ′′′ , . . . , t ′′′ τ ) of ( t ′′ , t ′′ , . . . , t ′′ τ ) such that(iii) for every α ∈ [ α ∗ t ′ ] and t, t ′ ∈ { t ′′′ , t ′′′ , . . . , t ′′′ τ } , V ( M ( t ) α ) = V ( M ( t ′ ) α ) .Since ( t ′′′ , t ′′′ , . . . , t ′′′ τ ) is a substring of ( t ′′ , t ′′ , . . . , t ′′ τ ) , ( t ′′′ , t ′′′ , . . . , t ′′′ τ ) satisfies (i)-(iii).Let τ = τ f ( η ) w +1 . Since ( t ′′′ , t ′′′ , . . . , t ′′′ τ ) satisfies (iii), for every α ∈ [ α ∗ t ′ ] and β <β ′ τ , the sum of the entries of a ( t ′′′ β ′ ) α is at most the sum of the entries of a ( t ′′′ β ) α . ByClaim 4.12.30, the sum of the entries of a ( t ′′′ β ) α is at most f ( η ) w . So there exists a substring ( q , q , q , . . . , q τ − ) of ( t ′′′ , t ′′′ , . . . , t ′′′ τ ) such that100iv) for every α ∈ [ α ∗ t ′ ] and t, t ′ ∈ { q , q , . . . , q τ − } , V ( M ( t ) α ) = V ( M ( t ′ ) α ) and a ( t ) α = a ( t ′ ) α .Since ( q , q , . . . , q τ − ) is a substring of ( t ′′′ , t ′′′ , . . . , t ′′′ τ ) , ( q , q , . . . , q τ − ) satisfies (i)-(iv).Since ( q , q , . . . , q τ − ) satisfies (i) and (iv), there exists β ∗ ∈ { , } such that α ∗ q β ∗ > α ∗ t ′ + 1 . For each γ ∈ [ τ − , redefine q γ to be q β ∗ + γ − . Then ( q , q , . . . , q τ − ) satisfies(i)-(iv) and α ∗ q > α ∗ t ′ + 1 . For each β ∈ [2 , τ − , by taking ( t , t , α ∗ , α ) = ( q , q β , α ∗ q , α ∗ t ′ ) in Claim 4.12.54, since ( q , . . . , q τ − ) satisfies (iv), there exists γ ∗ β ∈ [ α ∗ t ′ + 1 , α ∗ q ] such that a ( q β ) γ ∗ β is a nonzero sequence. For each β ∈ [2 , τ − , we choose γ ∗ β = α ∗ q if possible.Let τ = ( τ − / . By Claim 4.12.32, if γ ∗ τ = α ∗ q , then a ( q β ) γ ∗ τ is a nonzero sequencefor every β ∈ [ τ ] . Since we choose γ ∗ β = α ∗ q for each β ∈ [2 , τ − if possible, we know if γ ∗ τ = α ∗ q , then γ ∗ β = α ∗ q for every β ∈ [ τ , τ − by Claim 4.12.32. Hence there exists asubstring ( q ′ , q ′ , . . . , q ′ τ ) of ( q , q , . . . , q τ − ) such that(v) there exists γ ∗ ∈ [ α ∗ t ′ + 1 , α ∗ q ] such that a ( t ) γ ∗ is a nonzero sequence for every t ∈{ q ′ , q ′ , . . . , q ′ τ } .Since ( q ′ , q ′ , . . . , q ′ τ ) is a substring of ( q , q , . . . , q τ − ) , ( q ′ , q ′ , . . . , q ′ τ ) satisfies (i)-(v). Notethat α ∗ q ′ > γ ∗ .Let τ = τ w · ( φ ( w )+1) · ( f ( η ) w +1) . By Claims 4.12.30, 4.12.32 and 4.12.51, there exists asubstring ( q ′′ , q ′′ , . . . , q ′′ τ ) of ( q ′ , q ′ , . . . , q ′ τ ) satisfying (i)-(v) and(vi) for every α ∈ [ α ∗ q ′ ] and t, t ′ ∈ { q ′′ , q ′′ , . . . , q ′′ τ } , V ( M ( t ) α ) = V ( M ( t ′ ) α ) and a ( t ) α = a ( t ′ ) α .Since ( q ′′ , q ′′ , . . . , q ′′ τ ) satisifies (i) and (vi), there exists β ∗ ∈ { , } such that a ∗ q ′′ β ∗ > α ∗ q ′ + 1 > γ ∗ + 1 . For every γ ∈ [ τ − , redefine q ′′ γ = q ′′ γ − β ∗ . So there exists the smallest integer ξ ∗ ∈ [ γ ∗ + 1 , | V ( G ) | ] such that a ( q ′′ ) ξ ∗ is a nonzero sequence. By (vi) and Claim 4.12.32, forevery α ∈ [ ξ ∗ − , a ( q ′′ k ) α is identical for every k ∈ [ τ − .Let Z = { γ ∈ [ α ∗ ] : a (1) γ is a nonzero sequence } . So | Z | = n . Let Z ′ = { γ ∈ [ ξ ∗ ] : a ( q ′′ ) γ isa nonzero sequence } . Let n ′ = | Z ′ | . Recall that for every α ∈ [ α ∗ − , a (1) α = a (2) α = · · · = a ( κ ) α .Since ξ ∗ > γ ∗ + 1 > α ∗ + 1 , Z − { α ∗ } ⊆ Z ′ . By (v), { γ ∗ , ξ ∗ } ⊆ Z ′ − Z . So n ′ > n + 1 . Inparticular, w − n > . By the minimality of w − n , τ − h ( w − n ′ ) h ( w − n − .But τ − > h ( w − n − 1) + 1 since κ > h ( w − n ) + 1 , a contradiction. This proves theclaim. Claim 4.12.56. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∈ S ◦ j .Let τ ∈ N . Let t , t , . . . , t τ be elements in K ∗ ( M ) such that t i +1 ∈ V ( T t i ) − { t i } for every i ∈ [ τ − . Then τ η .Proof. Suppose τ > η + 1 . For each α ∈ [ τ ] , let ( a ( α )1 , a ( α )2 , . . . , a ( α ) | V ( G ) | ) be the ( i t α , j, M ) -truncation. Since η + 1 > , by Claim 4.12.52, there exists γ ∗ such that a (1) γ ∗ is a nonzerosequence. We may assume that γ ∗ is as small as possible. So a (1) γ is a zero sequence for every γ ∈ [ γ ∗ − . By Claim 4.12.32, for every γ ∈ [ γ ∗ − and α ∈ [ τ ] , a ( α ) γ is a zero sequence. ByClaim 4.12.55, τ h ( w − . So η h ( w − − , a contradiction.101or j ∈ [ |V| − and a c -monochromatic component M with S M ∈ S ◦ j , we define T M to bethe rooted tree obtained from T r M by contracting each maximal subtree of T M rooted at a nodein K ∗ ( M ) containing exactly one node in K ∗ ( M ) into a node. Note that | V ( T M ) | = | K ∗ ( M ) | . Claim 4.12.57. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∈ S ◦ j .If P is a directed path in T M , then | V ( P ) | η .Proof. By the definition of T M , there exist nodes t , t , . . . , t | V ( P ) | in K ∗ ( M ) such that foreach α ∈ [ | V ( P ) | − , t α +1 ∈ V ( T t α ) − { t α } . By Claim 4.12.56, | V ( P ) | η . Claim 4.12.58. Let j ∈ [ |V| − . Let M be a c -monochromatic component with S M ∈ S ◦ j .Then the maximum degree of T M is at most f ( η ) .Proof. For every node t ∈ K ∗ ( M ) − { r M } , let q t be the node in K ( M ) such that t ∈ ∂T j,q t .Let x be a node of T M . It suffices to show that the degree of x in T M is at most f ( η ) .Let R x be the subtree of T r M in T such that x is obtained by contracting R x . We say anode q in R x is important if q = q t for some t ∈ K ∗ ( M ) − { r M } . Note that the degree of x equals the number of nodes t in K ∗ ( M ) − { r M } such that there exists an important node q in R x with q = q t . Let r x be the root of R x . Since r x is the only element in V ( R x ) ∩ K ∗ ( M ) , forevery important node q , V ( M ) ∩ Y ( i q , | V ( T ) | +1 ,s +2) ∩ X V ( T q ) = V ( M ) ∩ Y ( i rx , | V ( T ) | +1 ,s +2) ∩ X V ( T q ) .Suppose there exists an important node q ∗ and node t ∗ ∈ K ∗ ( M ) −{ r M } with q ∗ = q t ∗ suchthat A L ( iq ∗ , | V ( T ) | +1 ,s +2) ( Y ( i q ∗ , | V ( T ) | +1 ,s +2) ∩ V ( M )) ∩ X V ( T t ∗ ) − X t ∗ = ∅ . Since t ∗ ∈ K ∗ ( M ) −{ r M } ,there exists v ∈ V ( M ) ∩ ( Y ( i t ∗ , | V ( T ) | +1 ,s +2) − Y ( i q ∗ , | V ( T ) | +1 ,s +2) ) ∩ X V ( T t ∗ ) . By Claim 4.12.25,since V ( M ) ∩ X q ∗ = ∅ and X q ∗ ∩ I j ⊆ W ( i q ∗ , − and t ∗ ∈ ∂T j,q ∗ , we have v X t ∗ . Since M isconnected, there exists a path P in M from v to V ( M ) ∩ Y ( i q ∗ , | V ( T ) | +1 ,s +2) internally disjointfrom Y ( i q ∗ , | V ( T ) | +1 ,s +2) . Let u be the neighbor of the end of P in V ( M ) ∩ Y ( i q ∗ , | V ( T ) | +1 ,s +2) .So u ∈ A L ( iq ∗ , | V ( T ) | +1 ,s +2) ( V ( M ) ∩ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ) . Since A L ( iq ∗ , | V ( T ) | +1 ,s +2) ( Y ( i q ∗ , | V ( T ) | +1 ,s +2) ∩ V ( M )) ∩ X V ( T t ∗ ) − X t ∗ = ∅ , u X V ( T t ∗ ) − X t ∗ . Since u ∈ V ( M ) and t ∗ ∈ ∂T j,q ∗ , q X t ∗ byClaim 4.12.25. Since v ∈ X V ( T t ∗ ) − X t ∗ , some internal node of P other than u belongs to X t ∗ .But V ( P ) ∩ X t ∗ ⊆ V ( M ) ∩ X t ∗ ⊆ V ( M ) ∩ X V ( T j,q ∗ ) = V ( M ) ∩ Y ( i q ∗ , | V ( T ) | +1 ,s +2) ∩ X V ( T j,q ∗ ) byClaim 4.12.25. Hence P is not internally disjoint from Y ( i q ∗ , | V ( T ) | +1 ,s +2) , a contradiction.Therefore, for every important node q and node t ∈ K ∗ ( M ) − { r M } with q = q t , we have A L ( iq, | V ( T ) | +1 ,s +2) ( Y ( i q , | V ( T ) | +1 ,s +2) ∩ V ( M )) ∩ X V ( T t ) − X t = ∅ . Hence for every t ∈ K ∗ ( M ) −{ r M } in which there exists an important node q in R x with q = q t , A L ( irx, | V ( T ) | +1 ,s +2) ( Y ( i rx , | V ( T ) | +1 ,s +2) ∩ V ( M )) ∩ X V ( T t ) − X t = A L ( irx, | V ( T ) | +1 ,s +2) ( Y ( i rx , | V ( T ) | +1 ,s +2) ∩ V ( M ) ∩ X V ( T q ) ) ∩ X V ( T t ) − X t = A L ( irx, | V ( T ) | +1 ,s +2) ( Y ( i q , | V ( T ) | +1 ,s +2) ∩ V ( M ) ∩ X V ( T q ) ) ∩ X V ( T t ) − X t ⊇ A L ( iq, | V ( T ) | +1 ,s +2) ( Y ( i q , | V ( T ) | +1 ,s +2) ∩ V ( M ) ∩ X V ( T t ) ) ∩ X V ( T t ) − X t = A L ( iq, | V ( T ) | +1 ,s +2) ( Y ( i q , | V ( T ) | +1 ,s +2) ∩ V ( M )) ∩ X V ( T t ) − X t = ∅ . Note that | A L ( irx, | V ( T ) | +1 ,s +2) ( Y ( i rx , | V ( T ) | +1 ,s +2) ∩ V ( M )) ∩ X V ( T t ) − X t | | A L ( irx, | V ( T ) | +1 ,s +2) ( Y ( i rx , | V ( T ) | +1 ,s +2) ∩ V ( M )) ∩ X V ( T rx ) − X r x | N > s ( | Y ( i rx , | V ( T ) | +1 ,s +2) ∩ V ( M ) ∩ X V ( T rx ) | ) f ( | Y ( i rx , | V ( T ) | +1 ,s +2) ∩ V ( M ) ∩ X V ( T rx ) | ) f ( η ) by Claim 4.12.24. So there are at most f ( η ) nodes t ∈ K ∗ ( M ) − { r M } in which there existsan important node q in R x with q = q t . Therefore, the degree of x is at most f ( η ) . Claim 4.12.59. If M is a c -monochromatic component with V ( M ) ∩ S |V|− j =1 I ◦ j = ∅ , then | V ( M ) | η .Proof. Since V ( M ) ∩ S |V|− j =1 I ◦ j = ∅ , there exists j ∈ [ |V| − such that S M ∈ S ◦ j . ByClaims 4.12.57 and 4.12.58, T M is a rooted tree with maximum degree at most f ( η ) and thelongest directed path in T M from the root contains at most η nodes. So | K ∗ ( M ) | = | V ( T M ) | P η − k =0 ( f ( η )) k ( f ( η )) η . By Claim 4.12.26, | V ( M ) | | K ∗ ( M ) |· η η · ( f ( η )) η = η .Now we are ready to complete the proof of this lemma. Let M be a c -monochromaticcomponent. If S M ∩ I ◦ j = ∅ for every j ∈ [ |V| − , then | V ( M ) | η λ by Claim 4.12.21.If S M ∩ I ◦ j = ∅ for some j ∈ [ |V| − , then | V ( M ) | η λ by Claim 4.12.59. Therefore, | V ( M ) | λ . This proves Lemma 4.12. The following lemma implies Theorem 1.19 by taking k = 1 and Y = ∅ . Lemma 4.13. For all s, t, w, k, ξ ∈ N , there exists λ ∈ N such that for every graph G containing no K s.t -subgraph, if Z is a subset of V ( G ) with | Z | ξ , V = ( V , V , . . . , V |V| ) isa Z -layering of G , ( T, X ) is a tree-decomposition of G − Z with V -width at most w , Y is asubset of V ( G ) with | Y | k , L is an ( s, V ) -compatible list-assignment of G such that ( Y , L ) is a V -standard pair, then there exists an L -coloring c of G such that every c -monochromaticcomponent contains at most λ vertices.Proof. Let s, t, w, k, ξ ∈ N . Let f be the function f s,t in Lemma 3.5. Let h be the identityfunction. For i ∈ N , let h i be the function defined by h i ( x ) := x + f ( h i − ( x )) for every x ∈ N .Let η := h s +1 ( k + ξ ) . Let η be the number λ in Lemma 4.11 taking s = s , t = t , w = w + ξ and η = h s +2 ( k + ξ ) + ( s + 1) ξ . Define λ := max { η , η } .Let G be a graph with no K s,t -subgraph, Z a subset of V ( G ) with | Z | ξ , V =( V , V , . . . , V |V| ) a Z -layering of G , ( T, X ) a tree-decomposition of G − Z with V -width atmost w , Y a subset of V ( G ) with | Y | k , L an ( s, V ) -compatible list-assignment of G suchthat ( Y , L ) is a V -standard pair. Say X = ( X p : p ∈ V ( T )) .We may assume that |V| > , since we may add empty layers into V . Let ( Y (0) , L (0) ) be a ( Z ∪ Y , -progress of ( Y , L ) . For each i ∈ [ s + 2] , define ( Y ( i ) , L ( i ) ) to be an ( N > sG ( Y ( i − ) , i ) -progress of ( Y ( i − , L ( i − ) . Note that every L ( s +2) -coloring of G is an L -coloring of G . It isclear that | Y ( i )1 | h i ( | Z ∪ Y | ) h i ( ξ + k ) for every i ∈ [0 , s + 2] by Lemma 3.5 and inductionon i . Claim 4.13.1. For every L ( s +2) -coloring of G , every monochromatic component intersecting Z ∪ Y is contained in G [ Y ( s +1)1 ] . roof. We shall prove that for each i ∈ [ s +2] and for every L ( i ) -coloring of G , every monochro-matic component colored i and intersecting Z ∪ Y is contained in G [ Y ( i − ] .Suppose to the contrary that there exist i ∈ [ s + 2] , an L ( i ) -coloring of G , and a monochro-matic component M colored i and intersecting Z ∪ Y such that V ( M ) Y ( i − . Since M intersects Z ∪ Y ⊆ Y ( i − , we have V ( M ) ∩ Y ( i − = ∅ . So there exist y ∈ V ( M ) ∩ Y ( i − and v ∈ N G ( y ) ∩ V ( M ) − Y ( i − . In particular, L ( i − ( y ) = { i } ⊆ L ( i ) ( v ) ⊆ L ( i − ( v ) .Since ( Y ( i − , L ( i − ) is a V -standard pair, (L2) implies that v N For all s, t, k ∈ N , there exist a number η > k and a nondecreasing function g with domain N and with g (0) > η such that if G is a graph with no K s,t -subgraph, r ∈ N , Y is a subset of V ( G ) with | Y | η , F is a set of colors with | F | r − , and L is an ( s, r, Y ) -list-assignment of G such that { y ∈ Y : x ∈ L ( y ) } is a stable set in G for every x ∈ F , then one of the following holds:1. There exists an ( η, g ) -bounded L -coloring of G such that for every x ∈ F , the set ofvertices colored x is a stable set in G .2. | Y | > k .3. For every color ℓ , there exist a subset Y ′ of V ( G ) with η > | Y ′ | > | Y | and an ( s, r, Y ′ ) -list-assignment L ′ of G with L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) , such that:(a) there does not exist an ( η, g ) -bounded L ′ -coloring of G such that for every x ∈ F ,the set of vertices colored x is a stable set in G ,(b) for every L ′ -coloring of G , every monochromatic component intersecting Y is con-tained in G [ Y ′ ] ,(c) { v ∈ Y : F ∩ L ( v ) = ∅} = { v ∈ Y ′ : F ∩ L ′ ( v ) = ∅} ,(d) for every x ∈ F ∪ { ℓ } and y ∈ Y ′ with x ∈ L ′ ( y ) , we have { v ∈ N G ( y ) − Y ′ : x ∈ L ′ ( v ) } = ∅ , and(e) for every v ∈ V ( G ) − Y ′ , we have L ′ ( v ) ∩ F = L ( v ) ∩ F .4. Y = ∅ , N G ( Y ) = ∅ , and there does not exist an ( η, g ) -bounded L | G − Y -coloring of G − Y such that for every x ∈ F , the set of vertices colored x is a stable set in G − Y .Proof. Let f be the function f s,t in Lemma 3.5. Let h − : N → N be the identity function,and let h : N → N be the function defined by h ( x ) := x + f ( x ) for x ∈ N . For i > , let h i : N → N be the function defined by h i ( x ) := h ( h i − ( x )) for x ∈ N . Let η := max { h k ( k ) , k + 1 } . Let g : N → N to be the function defined by g ( x ) := h x ( x ) + η for x ∈ N .Suppose that Statements 1, 2 and 4 do not hold. Suppose to the contrary that Statement 3does not hold for some color ℓ .First suppose that Y = ∅ . Define R to be a ( { v } , F ∪ { ℓ } ) -progress of L , where v is avertex of G . If G has an ( η, g ) -bounded R -coloring such that for every x ∈ F , the set ofvertices colored x is a stable set, then G has an ( η, g ) -bounded L -coloring such that for every x ∈ F , the set of vertices colored x is a stable set, so Statement 1 holds, a contradiction. So G has no ( η, g ) -bounded R -coloring such that for every x ∈ F , the set of vertices colored x isa stable set, then Statement 3 holds if we take Y ′ = { v } and L ′ = R by Lemma 4.4.Hence Y = ∅ . Suppose that N G ( Y ) = ∅ . If V ( G ) = Y , then there exists an ( η, g ) -bounded L -coloring of G such that for every x ∈ F , the set of vertices colored x is a stableset, a contradiction. So V ( G ) = Y . Define G ′ = G − Y . Since Statement 4 does not hold,there exists an ( η, g ) -bounded L | G ′ -coloring of G ′ such that for every x ∈ F , the set of verticescolored x is a stable set. We can further color the vertices in Y by the unique element in106heir lists to obtain an ( η, g ) -bounded L -coloring of G such that for every x ∈ F , the set ofvertices colored x is a stable set, a contradiction.Hence Y = ∅ and N G ( Y ) = ∅ . Let Y = N For every L ∗ -coloring c of G , every c -monochromatic component intersecting Y is contained in G [ Y ∗ ] .Proof. For i ∈ [ | Y | ] , let M i be the c -monochromatic component containing y i . We shall provethat V ( M i ) ⊆ U i − for every i ∈ [ | Y | ] .For i ∈ [ | Y | ] , note that U i − U i − = N > s ( U i − ) . Since L i is an ( U i − U i − , { ℓ i }∪ F ) -progress, ℓ i L i ( u ) for every u ∈ U i − U i − , and for every v ∈ N For all s, t, θ, η, r ∈ N with η > θ + 1 , for every nondecreasing function g withdomain N , if G is a graph with no K s,t -subgraph, Y is a subset of V ( G ) with θ + 1 | Y | η , F is a set of colors with | F | r , and L is an ( s, r, Y ) -list-assignment of G such that { y ∈ Y : x ∈ L ( y ) } is a stable set in G for every x ∈ F , then at least one of the followingholds:1. There exists an ( η, g ) -bounded L -coloring of G such that for every x ∈ F , the set ofvertices colored x is a stable set in G .2. There exist an induced subgraph G ′ of G with | V ( G ′ ) | < | V ( G ) | , a subset Y ′ of V ( G ′ ) with | Y ′ | η , and an ( s, r, Y ′ ) -list-assignment L ′ of G ′ such that the following hold:(a) L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ′ ) .(b) There does not exist an ( η, g ) -bounded L ′ -coloring of G ′ such that for every x ∈ F ,the set of vertices colored x is a stable set in G ′ .(c) { v ∈ Y : F ∩ L ( v ) = ∅} ∩ V ( G ′ ) = { v ∈ Y ′ : F ∩ L ′ ( v ) = ∅} .(d) For every v ∈ V ( G ′ ) − Y ′ , we have L ′ ( v ) ∩ F = L ( v ) ∩ F .3. T := { ( A, B ) : | V ( A ∩ B ) | < θ, | V ( A ) ∩ Y | θ } is a tangle of order θ in G .Proof. Suppose that Statements 1, 2 and 3 do not hold. Since T is not a tangle, one of (T1),(T2) or (T3) is violated.Suppose that (T2) violated. So there exist ( A i , B i ) ∈ T for i ∈ [3] such that A ∪ A ∪ A = G . Hence | Y | P i =1 | A i ∩ Y | θ , a contradiction. So T satisfies (T2). Similarly, for every ( A, B ) ∈ T , we have V ( A ) = V ( G ) ; otherwise | Y | = | V ( A ) ∩ Y | θ , a contradiction. So T satisfies (T3).Therefore, (T1) is violated. So there exists a separation ( A, B ) of G of order less than θ such that ( A, B ) 6∈ T and ( B, A ) 6∈ T . That is, | V ( A ) ∩ Y | > θ and | V ( B ) ∩ Y | > θ . Inparticular, V ( A ) = V ( G ) and V ( B ) = V ( G ) .Let Y A := ( Y ∩ V ( A )) ∪ V ( A ∩ B ) and Y B := ( Y ∩ V ( B )) ∪ V ( A ∩ B ) . Note that max {| Y A | , | Y B |} | Y |− θ < η . Let G A := G [ V ( A )] and G B := G [ V ( B )] . Let Z A := N For every z ∈ Z A , there exists a subset L A ( z ) of L ( z ) − S { L ( v ) : v ∈ N G A ( z ) ∩ Y A } with size s + r − | N G A ( z ) ∩ Y A | such that L A ( z ) ∩ F = L ( z ) ∩ F and L A ( z ) ∩ L A ( u ) = ∅ for every u ∈ N G ( z ) ∩ Y A .Proof. Note that z ∈ V ( A ) − V ( B ) since V ( A ∩ B ) ⊆ Y A . So N G A ( z ) ∩ Y = N G ( z ) ∩ Y .Hence | N G A ( z ) ∩ Y A | > | N G ( z ) ∩ Y | . Since z ∈ Z A , we have | N G ( z ) ∩ Y | s − . So either z ∈ N For all s, t, w ∈ N there exists η ∈ N and a nondecreasing function g suchthat if G is a graph of treewidth at most w and with no K s,t -subgraph, Y is a subset of V ( G ) with | Y | η , and L is an ( s, , Y ) -list-assignment of G , then there exists an ( η, g ) -bounded L -coloring of G .Proof. Define η and g to be the number η and the function g in Lemma 5.1 by taking s = s , t = t , k = 9 w + 18 and r = 1 . Note that g ( x ) > η > w + 18 for every x ∈ N by Lemma 5.1.Suppose to the contrary that this theorem does not hold. So there exist a graph oftreewidth at most w and with no K s,t -subgraph, a subset Y of V ( G ) with | Y | η , and an ( s, , Y ) -list-assignment L of G such that there does not exist an ( η, g ) -bounded L -coloringof G . Assume further that | V ( G ) | is as small as possible and subject to this, | Y | is as large aspossible. Since g ( x ) > η for every x ∈ N , we have | V ( G ) | > η , as otherwise any L -coloringof G is ( η, g ) -bounded.By Lemma 5.1 and the choice of G and Y , we have | Y | > w + 18 . By Lemma 5.2, thereexists a tangle of order w + 2 in G . But G has treewidth at most w , there exists no tangle oforder w + 2 in G by [64, Lemma (5.2)], a contradiction. This proves the theorem.110 orollary 5.4. For all s, t, w ∈ N , there exists η ∈ N and a nondecreasing function g withdomain N such that if G is a graph of treewidth at most w and with no K s,t -subgraph and L is an ( s + 1) -list-assignment of G , then there exists an L -coloring of G with clustering η g ( η ) .Proof. Define L ′ ( v ) to be an ( s + 1) -element subset of L ( v ) for every v ∈ V ( G ) . Clearly, L ′ is an ( s, , ∅ ) -list-assignment of G , and every L ′ -coloring is an L -coloring. The resultimmediately follows from Theorem 5.3.Observe that Corollary 5.4 implies Theorem 1.14. Our proofs of Theorems 1.2 and 1.16 depend on the Graph Minor Structure Theorem,which we now introduce. Recall the definition of an H -minor α in a graph G from Section 2.A tangle T in G controls an H -minor α if there does not exist ( A, B ) ∈ T of order less than | V ( H ) | such that V ( α ( h )) ⊆ V ( A ) for some h ∈ V ( H ) .A society is a pair ( S, Ω) , where S is a graph and Ω is a cyclic permutation of a subset Ω of V ( S ) . For ρ ∈ N , a society ( S, Ω) is a ρ -vortex if for all distinct u, v ∈ Ω , there do notexist ρ + 1 disjoint paths in S between I ∪ { u } and J ∪ { v } , where I is the set of vertices in Ω after u and before v in the order Ω , and J is the set of vertices in Ω after v and before u . Fora society ( S, Ω) with Ω = ( v , v , . . . , v n ) in order, a vortical decomposition of ( S, Ω) is a path-decomposition ( t t · · · t n , X ) such that the i -th bag X i of X contains the i -th vertex v i for each i ∈ [ n ] . The adhesion of such a vortical decomposition is max {| X i ∩ X j | : i, j ∈ [ n ] , i = j } .We use the following theorem of Robertson and Seymour [63]. Theorem 5.5 ([63, (8.1)]) . Every ρ -vortex has a vortical decomposition of adhesion at most ρ . A segregation of a graph G is a set S of societies such that: • S is a subgraph of G for every ( S, Ω) ∈ S , and S { S : ( S, Ω) ∈ S} = G , and • for all distinct ( S, Ω) and ( S ′ , Ω ′ ) ∈ S , we have V ( S ∩ S ′ ) ⊆ Ω ∩ Ω ′ and E ( S ∩ S ′ ) = ∅ .We write V ( S ) = S { Ω : ( S, Ω) ∈ S} . For positive integers κ, ρ , a segregation S is of type ( κ, ρ ) if there exist disjoint subsets S , S of S with S = S ∪ S and |S | κ such that | Ω | for every ( S, Ω) ∈ S , and every member of S is a ρ -vortex. For a tangle T in G ,a segregation S of G is T -central if for every ( S, Ω) ∈ S , there exists no ( A, B ) ∈ T with B ⊆ S .Let Σ be a surface. For every subset ∆ of Σ , we denote the closure of ∆ by ∆ and theboundary of ∆ by ∂ ∆ . An arrangement of a segregation S = { ( S , Ω ) , . . . , ( S k , Ω k ) } in Σ isa function α with domain S ∪ V ( S ) , such that: • For [ k ] , α ( S i , Ω i ) is a closed disk ∆ i ⊆ Σ , and α ( x ) ∈ ∂ ∆ i for each x ∈ Ω i . • For i, j ∈ [ k ] with i = j , if x ∈ ∆ i ∩ ∆ j , then x = α ( v ) for some v ∈ Ω i ∩ Ω j . • For all distinct x, y ∈ V ( S ) , we have α ( x ) = α ( y ) . • For i ∈ [ k ] , Ω i is mapped by α to a natural order of α (Ω i ) determined by ∂ ∆ i .111n arrangement is proper if ∆ i ∩ ∆ j = ∅ whenever | Ω i | , | Ω j | > , for all i < j k .For a tangle T in a graph G of order θ and a subset Z of V ( G ) with | Z | < θ , T − Z isdefined to be the set of all separations ( A ′ , B ′ ) of G − Z of order less than θ − | Z | such thatthere exists ( A, B ) ∈ T with Z ⊆ V ( A ∩ B ) , A ′ = A − Z and B ′ = B − Z . It is proved inRobertson and Seymour [64] that T − Z is a tangle in G − Z of order θ − | Z | .The following is the Graph Minor Structure Theorem of Robertson and Seymour [65]. Theorem 5.6 ([65, (3.1)]) . For every graph H , there exist κ, ρ, ξ, θ ∈ N such that if T is atangle of order at least θ in a graph G controlling no H -minor of G , then there exist Z ⊆ V ( G ) with | Z | ξ and a ( T − Z ) -central segregation S of G − Z of type ( κ, ρ ) such that S has aproper arrangement in some surface in which H cannot be embedded. The goal of this section is to prove Theorems 1.2 and 1.16. Let G be a graph. A location L in G is a collection of separations of G such that A ⊆ B ′ for every ordered pair of distinctmembers ( A, B ) , ( A ′ , B ′ ) of L . Define G ( L ) to be the graph G [ T ( A,B ) ∈L V ( B )] .Let s ∈ N , r ∈ N and ℓ ∈ [0 , s +2] . Let G be a graph and let Y ⊆ V ( G ) . A list-assignment L of G is an ( s, Y , ℓ, r ) -list-assignment if the following hold:(R1) L ( v ) ⊆ [ s + 2 + r ] for all v ∈ V ( G ) .(R2) L is an ( s, r + 2 , Y ) -list-assignment of G .(R3) For every y ∈ Y and color x ∈ { ℓ } ∪ [ s + 3 , s + 2 + r ] with x ∈ L ( y ) , we have { v ∈ N G ( y ) − Y : x ∈ L ( v ) } = ∅ .(R4) For every x ∈ [ s + 3 , s + 2 + r ] , the set { y ∈ Y : x ∈ L ( y ) } is a stable set in G .(R5) For every v ∈ V ( G ) − Y , |{ y ∈ N G ( v ) ∩ Y : L ( y ) ⊆ [ s + 3 , s + r + 2] }| = r − | L ( v ) ∩ [ s + 3 , s + 2 + r ] | . Note that an ( s, Y , , -list-assignment is the same as a restricted ( s, , Y ) -list-assignment,which we use for graph minors. The more general setting of ( s, Y , ℓ, r ) -list-assignments areused for odd minors.Let G be a graph and Z, Y ⊆ V ( G ) . Let s ∈ N , r ∈ N and ℓ ∈ [0 , s + 2] . Let L be an ( s, Y , ℓ, r ) -list-assignment of G . A ( Z, ℓ ) -growth of L is a list-assignment L ′ of G defined asfollows: • Let Y ( − := Y and L ( − ( v ) := L ( v ) for every v ∈ V ( G ) . • Let U := Z , and for each i > , let U i := N > s ( Y ( i − ) . • Let Y ( i )1 := Y ( i − ∪ U i for i > . • For every i > , let L ( i ) be a ( U i , { ℓ, i } ∪ [ s + 3 , s + 2 + r ]) -progress of L ( i − such that L ( i ) ( v ) ∩ [ s + 3 , s + 2 + r ] = L ( i − ( v ) ∩ [ s + 3 , s + 2 + r ] for every v ∈ V ( G ) − ( Y ( i − ∪ U i ) .(Note that such an L ( i ) exists since s + 2 > s − and for every v ∈ V ( G ) − ( Y ( i − ∪ U i ) , L ( i ) ( v ) can be obtained from L ( i − ( v ) by removing elements, and L ( i ) ( v ) = L ( i − ( v ) only when v ∈ N S,X ) be the separation of G such that V ( A S,X ) := X ∪ Z and V ( A S,X ∩ B S,X ) := ∂X ∪ Z , and subject to these conditions, E ( A S,X ) is minimal. Let L := { ( A S,X , B S,X ) :( S, Ω) ∈ S , X ∈ X S } and L := L ∪ L . Note that every member of L has order at most ξ + 3 + 2 ρ < θ ∗ θ . Since S is ( T − Z ) -central, L ⊆ T . In addition, L is a location with Z ⊆ V ( A ∩ B ) for every ( A, B ) ∈ T .Let L ′ be a ( Z, ℓ ) -growth of L , and let Y ′ := { v ∈ V ( G ) : | L ′ ( v ) | = 1 } . By Lemma 5.7, | Y ′ ∩ V ( A ) | h ( | V ( A ∩ B ) | + | Y ∩ V ( A ) | ) h (4 θ ) for every ( A, B ) ∈ L . For each ( A, B ) ∈ L ,define: • U (0) A := ( Y ′ ∩ V ( A )) ∪ V ( A ∩ B ) , • U ( i ) A := N > sG [ V ( A )] ( S i − j =0 U ( j ) A ) for each i > , and • Z A := S s +3 i =0 U ( i ) A .Note that for every ( A, B ) ∈ L and i ∈ [0 , s + 3] , we have | S ij =0 U ( j ) A | f i ( h (4 θ ) + θ ) byLemma 3.5. So | Z A | f s +3 ( h (4 θ ) + θ ) .Let G ′ = G [( T ( A,B ) ∈L V ( B )) ∪ S ( A,B ) ∈L Z A ] . Since Y ′ ∩ V ( A ) ⊆ Z A for every ( A, B ) ∈ L ,we have Y ′ ⊆ V ( G ′ ) . Claim 5.8.1. There exists a Z -layering V = ( V , V , . . . , V |V| ) of G ′ and a tree-decompositionof G ′ with V -width at most w ( θ ) such that for every ( A, B ) ∈ L , there exists a positive integer i A such that V ( A ∩ B ) − Z ⊆ V i A − ∪ V i A ∪ V i A +1 , and U ( i ) A ⊆ V i A + i for every i ∈ [ s + 3] , and U (0) A − V ( A ∩ B ) ⊆ V i A , where V = ∅ .Proof. Let G be the graph obtained from G [ S ( S, Ω) ∈S Ω] by adding a new vertex v S adjacentto all vertices in Ω for each ( S, Ω) ∈ S . Since there exists a proper arrangement of S in Σ , G can be embedded in Σ . Since K t ′ cannot be embedded in Σ , the Euler genus of Σ is atmost σ . By [17, Theorem 12], there exists a layering V = ( V , , V , , . . . , V , |V | ) of G and atree-decomposition ( T , X ) of G of V -width at most σ + 3 . For each p ∈ V ( T ) , we denotethe bag at p in ( T , X ) by X ,p .Let G be the graph obtained from G [ T ( A,B ) ∈L V ( B )] − Z by adding a new vertex v A adjacent to all vertices in V ( A ∩ B ) − Z for each ( A, B ) ∈ L , and adding a new vertex v S adjacent to all vertices in Ω for each ( S, Ω) ∈ S . Note that V ( G ) ⊆ V ( G ) . For each i ∈ [ |V | ] , let V ,i := V ,i ∪ [ ( S, Ω) ∈S ,v S ∈ V ,i ( V ( G ) ∩ V ( S ) − Ω) . Let V = ( V , , V , , . . . , V , |V | ) . Since N G ( V ( G ) ∩ V ( S ) − Ω) ⊆ N G ( v S ) for every ( S, Ω) ∈ S ,we have that V is a layering of G . For each node p ∈ V ( T ) , define X ,p := X ,p ∪ [ ( S, Ω) ∈S [ X ∈X S ,X ∩ Ω ∈ X ,p ( X ∩ V ( G )) . ( T , X ) is a tree-decomposition of G and ( P S , X S ) is a vortical decomposition of S for every ( S, Ω) ∈ S , ( T , X ) is a tree-decomposition of G , where for every p ∈ V ( T ) , thebag of ( T , X ) at p is X ,p . Since ( P S , X S ) has adhesion at most ρ for each ( S, Ω) ∈ S , the V -width of ( T , X ) is at most (2 σ + 3)(2 ρ + 1) .For each ( A, B ) ∈ L , let i A be the index i such that v A ∈ V ,i ; for each ( A, B ) ∈ L , let i A be the index i such that v S ∈ V ,i , where ( S, Ω) is the member in S such that V ( A ) ⊆ V ( S ) . By adding empty layers, we may assume that |V | > s + 2 + max ( A,B ) ∈L i A . For each ( A, B ) ∈ L , since v A ∈ V ,i A and every vertex in V ( A ∩ B ) − Z is adjacent to v A , we have V ( A ∩ B ) − Z ⊆ V ,i A − ∪ V ,i A ∪ V ,i A +1 .For each j ∈ [ |V | ] , define V ,j := (cid:0) V ,j ∪ [ ( A,B ) ∈L [ i ∈ [ s +3] ,i A + i = j U ( i ) A ∪ [ ( A,B ) ∈L ,i A = j ( U (0) A − V ( G )) (cid:1) ∩ V ( G ′ ) − Z. Define V = ( V , , V , , . . . , V , |V | ) . Then V is a Z -layering of G ′ such that U ( i ) A ⊆ V ,i A + i forevery ( A, B ) ∈ L and i ∈ [ s + 3] . Since for every ( A, B ) ∈ L , we have ( U (0) A − V ( G )) ∩ V ( G ′ ) − Z = U (0) A − V ( A ∩ B ) , implying U (0) A − V ( A ∩ B ) ⊆ V ,i A . For each p ∈ V ( T ) , define X ,p := (cid:0) X ,p ∪ [ ( A,B ) ∈L ,v A ∈ X ,p ( Z A − V ( G )) ∪ [ ( A,B ) ∈L ,V ( A ∩ B ) − Z ⊆ X ,p ( Z A − V ( G )) (cid:1) ∩ V ( G ′ ) . Then ( T , X ) is a tree-decomposition of G ′ , where for every p ∈ V ( T ) , the bag of ( T , X ) at p is X ,p . Since | Z A | f s +3 ( h (4 θ ) + θ ) , the V -width of ( T , X ) is at most (2 σ + 3)(2 ρ + 1) · f s +3 ( h (4 θ ) + θ ) w ( θ ) .Then V and ( T , X ) are the desired Z -layering and tree-decomposition of G ′ , respectively.Let V := ( V , V , . . . , V |V| ) be the Z -layering of G ′ mentioned in Claim 5.8.1. Let L L bethe following list-assignment of G ′ : • For every v ∈ V ( G ′ ) − Y ′ , let L L ( v ) := L ′ ( v ) ∩ [ s + 2] − { i } , where i ∈ [ s + 2] is thenumber such that v ∈ V j and j ≡ i (mod s + 2 ). • For every v ∈ Y ′ , let L L ( v ) := L ′ ( v ) ∩ [ s + 2] .Let G ′′ be the subgraph of G ′ induced by { v ∈ V ( G ′ ) : L L ( v ) = ∅} . Then L L | G ′′ is an ( s, V ) -compatible list-assignment of G ′′ . Since L ′ satisfies (R5), for every v ∈ V ( G ′ ) − Y ′ , | L L ( v ) | > | L ′ ( v ) | − | L ′ ( v ) ∩ [ s + 3 , s + 2 + r ] | − | L ′ ( v ) | − ( r − |{ y ∈ N G ( v ) ∩ Y ′ : L ′ ( y ) ⊆ [ s + 3 , s + r + 2] }| ) − | L ′ ( v ) | − ( r − | N G ( v ) ∩ Y ′ − V ( G ′′ ) | ) − | L ′ ( v ) | − r + | N G ( v ) ∩ Y ′ | − | N G ( v ) ∩ Y ′ ∩ V ( G ′′ ) | − . Since L ′ is an ( s, r + 2 , Y ′ ) -list-assignment of G , the following hold: • For every y ∈ V ( G ′′ ) ∩ N For each ( A, B ) ∈ L , there exist Y A ⊆ V ( A ) with ( Y ′ ∪ Z A ) ∩ V ( A ) ⊆ Y A andwith | Y A | η ∗ ( θ ) and an ( s, Y A , ℓ, r ) -list-assignment L A of G [ V ( A )] such that • L A ( v ) = { c L ( v ) } for every v ∈ ( Y ′ ∪ Z A ) ∩ V ( A ) , • L A ( v ) ⊆ L ′ ( v ) for every v ∈ V ( A ) , and • for every L A -coloring of G [ V ( A )] , every monochromatic component intersecting ( Y ′ ∩ V ( A )) ∪ V ( A ∩ B ) is contained in G [ Z A ] .Proof. For every separation ( A, B ) , define the following: • Let Y ′ A := ( Y ′ ∪ Z A ) ∩ V ( A ) . • For every v ∈ V ( A ) ∩ Y ′ A , let L ′ A ( v ) := { c L ( v ) } . • For every v ∈ V ( A ) − Y ′ A with | N G [ V ( A )] ( v ) ∩ Y ′ A | ∈ [ s − , let L ′ A ( v ) be a subset of L ′ ( v ) with size s + r + 2 − | N G [ V ( A )] ( v ) ∩ Y ′ A | such that L ′ A ( v ) ∩ L ′ A ( y ) = ∅ for every y ∈ Y ′ A ,and |{ y ∈ N G ( v ) ∩ Y ′ A : L ′ A ( y ) ⊆ [ s + 3 , s + 2 + r ] }| = r − | L ′ A ( v ) ∩ [ s + 3 , s + 2 + r ] | .(Such L ′ A ( v ) exist since v ∈ V ( A ) − Y ′ A implies v ∈ V ( A ) − V ( B ) , and L ′ satisfies (R5).) • For every v ∈ V ( A ) − Y ′ A with | N G [ V ( A )] ( v ) ∩ Y ′ A | > s and with | N G [ V ( A )] ∩ ( Y ′ A − U ( s +3) A ) | ∈ [ s − , let L ′ A ( v ) be a subset of L ′ ( v ) with size s + r + 2 − | N G [ V ( A )] ∩ ( Y ′ A − U ( s +3) A ) | such that L ′ A ( v ) ∩ L ′ A ( y ) = ∅ for every y ∈ Y ′ A − U ( s +3) A , and |{ y ∈ N G ( v ) ∩ Y ′ A : L ′ A ( y ) ⊆ [ s + 3 , s + 2 + r ] }| = r − | L ′ A ( v ) ∩ [ s + 3 , s + 2 + r ] | . (Such L ′ A ( v ) exist since v ∈ V ( A ) − Y ′ A implies v ∈ V ( A ) − V ( B ) , and L ′ satisfies (R5).) • For every other vertex v in V ( A ) , let L ′ A ( v ) := L ′ ( v ) .Since L ′ is an ( s, r + 2 , Y ′ ) -list-assignment of G , we have L ′ A is an ( s, r + 2 , Y ′ A ) -list-assignmentof G [ V ( A )] . Since { v ∈ V ( G ′ ) : c L ( v ) = x } = { v ∈ Y ′ : L ′ ( v ) = { x }} is a stable set forevery x ∈ [ s + 3 , s + 2 + r ] , and L ′ satisfies (R3) and (R4) (with Y replaced by Y ′ ), we know117 ′ A satisfies (R4) (with Y replaced by Y ′ A ), and { v ∈ N G [ V ( A )] ( y ) − Y ′ A : x ∈ L ′ A ( v ) } = ∅ for every y ∈ Y ′ A and x ∈ [ s + 3 , s + 2 + r ] with x ∈ L ′ A ( y ) . Since L ′ satisfies (R5) (with Y replaced by Y ′ ), L ′ A satisfies (R5) (with Y replaced by Y ′ A ) by the definition of L ′ A . Inaddition, | Y ′ A | | Y ′ ∩ V ( A ) | + | Z A | h (4 θ ) + f s +3 ( h (4 θ ) + θ ) .Furthermore, define the following: • Let U := { v ∈ V ( A ) − Y ′ A : | N G [ V ( A )] ( v ) ∩ Y ′ A | > s } . • Let Y A := Y ′ A ∪ U . • Let L A be a ( U, { ℓ } ∪ [ s + 3 , s + 2 + r ]) -progress of L ′ A (in G [ A ] ).Since L ′ A is an ( s, r + 2 , Y ′ A ) -list-assignment of G [ V ( A )] , we have L A satisfies (R1)–(R3) (with Y replaced by Y A ) by Lemma 4.4. By the definition of L A , for every x ∈ [ s +3 , s +2+ r ] , we have { y ∈ Y A : x ∈ L A ( y ) } = { y ∈ Y ′ A : x ∈ L ′ A ( y ) } , so L A satisfies (R4) and (R5). Hence L A is an ( s, Y A , ℓ, r ) -list-assignment of G [ V ( A )] . Furthermore, it is clear that L A ( v ) ⊆ L ′ A ( v ) ⊆ L ′ ( v ) for every v ∈ V ( A ) , so L A ( v ) = { c L ( v ) } for every v ∈ Y ′ A , and every L A -coloring is a L ′ A -coloring.Note that for every ( A, B ) ∈ L , by Lemma 3.5, | Y A | | Y ′ A | + | U | h (4 θ ) + f s +3 ( h (4 θ ) + θ ) + f ( h (4 θ ) + f s +3 ( h (4 θ ) + θ )) = η ∗ ( θ ) . In addition, for every ( A, B ) ∈ L , we have Y ∩ V ( A ) ⊆ Y ′ A ⊆ Y A ⊆ V ( A ) .Now assume that ( A, B ) is a fixed element of L . We shall prove that for every L A -coloring c A of G [ V ( A )] , every c A -monochromatic component M intersecting ( Y ′ ∩ V ( A )) ∪ V ( A ∩ B ) is contained in G [ Z A ] . Suppose that M is not contained in G [ Z A ] .Since L A ( v ) ⊆ L ′ ( v ) for every v ∈ V ( A ) , c A is an L ′ | G [ V ( A )] -coloring of G [ V ( A )] . Wecan extend c A to be an L ′ -coloring c ′ of G by coloring each vertex v in V ( G ) − V ( A ) withan arbitrary color in L ′ ( v ) . So M is a connected subgraph of M ′ ∩ G [ A ] , where M ′ is a c ′ -monochromatic component intersecting ( Y ′ ∩ V ( A )) ∪ V ( A ∩ B ) . Since G [ Y ′ ∩ V ( A )] ⊆ G [ Z A ] ,if V ( M ) ∩ ( Y ∪ Z ) = ∅ , then M ′ intersects Y ∪ Z , so M ′ is contained in G [ Y ′ ] by Lemma 5.7,and hence V ( M ) ⊆ Y ′ ∩ V ( A ) ⊆ Z A , a contradiction. So V ( M ) ∩ ( Y ∪ Z ) = ∅ . Hence V ( M ) ∩ (( Y ′ ∩ V ( A ) − ( Y ∪ Z )) ∪ ( V ( A ∩ B ) − Y ′ )) = ∅ .Note that ( Y ′ ∩ V ( A ) − ( Y ∪ Z )) ∪ ( V ( A ∩ B ) − Y ′ ) ⊆ U (0) A ⊆ Z A . Since M intersects U (0) A ⊆ Z A and V ( M ) Z A , there exist v ∈ V ( M ) − Z A and y ∈ V ( M ) ∩ Z A such that y ∈ N G [ V ( A )] ( v ) ∩ Z A = ∅ and L ′ A ( v ) ∩ L ′ A ( y ) = ∅ . Since Z A ⊆ Y A and L A satisfies (R3),the vertices of M are colored with some color in [ s + 2] . Recall that by Claim 5.8.1, U (0) A ⊆ V i A − ∪ V i A ∪ V i A +1 and U ( s +3) A ⊆ V i A + s +3 . Since L L is ( s, V ) -compatible, no component of M [ V ( M ) ∩ Z A ] intersects both U (0) A and U ( s +3) A . So y, v can be chosen so that y ∈ V ( M ) ∩ ( S s +2 j =0 U ( j ) A ) and v ∈ N G ( y ) ∩ ( V ( M ) − Z A ) such that L ′ A ( v ) ∩ L ′ A ( y ) = ∅ . Since L ′ A satisfies(L3), we have | N G [ V ( A )] ( v ) ∩ Y ′ A | > s . Since y ∈ V ( M ) ∩ ( S s +2 j =0 U ( j ) A ) and L ′ A ( v ) ∩ L ′ A ( y ) = ∅ ,the definition of L ′ A ( v ) implies that | N G [ V ( A )] ( v ) ∩ ( Y ′ A − U ( s +3) A ) | > s . Since Y ′ ∩ V ( A ) ⊆ Z A ,we have Y ′ A − U ( s +3) A = S s +2 j =0 U ( j ) A . Hence | N G [ V ( A )] ( v ) ∩ S s +2 j =0 U ( j ) A | > s . So v ∈ U ( s +3) A ⊆ Z A , acontradiction. This shows that M is contained in G [ Z A ] and proves this claim.For every ( A, B ) ∈ L , let L A and Y A be the list-assignment and set mentioned inClaim 5.8.2, respectively, so | Y A | η ∗ ( θ ) . For every ( A, B ) ∈ L , since Y ∩ V ( A ) ⊆ Y A ⊆ V ( A ) ,118here exists an ( η, g ) -bounded L A -coloring c A of G [ V ( A )] such that for every x ∈ [ s +3 , s +2+ r ] ,the set of vertices colored x is a stable set by our assumption. Since L is a location, for every v ∈ V ( G ) − V ( G ′ ) , there uniquely exists ( A v , B v ) ∈ L such that v ∈ V ( A v ) − V ( B v ) . Let c ∗ be the following function: • For every v ∈ V ( G ′ ) , let c ∗ ( v ) := c L ( v ) . • For every v ∈ V ( G ) − V ( G ′ ) , let c ∗ ( v ) := c A v ( v ) .Clearly, c ∗ is an L ′ -coloring (and hence an L -coloring) of G . Suppose that there exists x ∗ ∈ [ s + 3 , s + 2 + r ] such that the set of vertices colored x ∗ is a not stable set. Then thereexists an edge e of G whose both ends are colored x ∗ . Since for every ( A, B ) ∈ L , the setof vertices colored x ∗ under c A is a stable set, e belongs to T ( A,B ) ∈L B ⊆ G ′ . But the set ofvertices colored x ∗ under c L is a stable set, a contradiction. So c ∗ is not ( η, g ) -bounded byour assumption.By Lemma 5.7, the union U ∗ of the c ∗ -monochromatic component intersecting Y ∪ Z iscontained in G [ Y ′ ] . Note that η > | Y | > θ . So U ∗ contains at most | Y ′ | h ( | Y | + ξ ) g ( | Y | ) | Y | g ( | Y | ) . Hence, there exists a c ∗ -monochromatic component M disjoint from Y ∪ Z containing more than η g ( η ) vertices.If V ( M ) ⊆ V ( G ′ ) , then M is a c L -monochromatic component in G ′ , so M contains at most η ( η + θ ) η (2 η ) g ( η ) η g ( η ) vertices, a contradiction. Hence V ( M ) V ( G ′ ) . So thereexists ( A, B ) ∈ L such that V ( M ) ∩ V ( A ) − ( V ( B ) ∪ Z A ) = ∅ . So V ( M ) ∩ V ( A ) − Z A = ∅ .If M [ V ( A )] is not connected, then every component of M [ V ( A )] is a c A -monochromaticcomponent intersecting V ( A ∩ B ) , so M [ V ( A )] ⊆ G [ Z A ] by Claim 5.8.2, a contradiction.So M [ V ( A )] is connected. Hence M is contained in G [ V ( A )] and is a c A -monochromaticcomponent. But c A is ( η, g ) -bounded, so M contains at most η g ( η ) vertices, a contradiction.This completes the proof.We now prove our main results for graphs excluding a minor or odd minor. We use thefollowing theorem of Geelen et al. [29]. Theorem 5.9 ([29, Theorem 13]) . There is a constant c such that for all ℓ ∈ N , if t := ⌈ cℓ √ log 12 ℓ ⌉ then for every graph G that contains a K t -minor α , either G contains an odd K ℓ -minor, or there exists a set X of vertices with | X | < ℓ such that the (unique) block U of G − X that intersects all branch sets of α disjoint from X is bipartite. Before proving Theorem 5.10 below, we note that it implies our main results from Section 1.In particular, the first part of Theorem 5.10 with Y = ∅ implies Theorem 1.2 since the list-assignment ( L ( v ) : v ∈ V ( G )) with L ( v ) = [ s + 2] for every v ∈ V ( G ) is an ( s, ∅ , , -list-assignment. The second part of Theorem 5.10 with Y = ∅ and ℓ = 1 implies Theorem 1.16since the list-assignment ( L ( v ) : v ∈ V ( G )) with L ( v ) = [2 s + 1] for every v ∈ V ( G ) is an ( s, ∅ , , s − -list-assignment. Theorem 5.10. For all s, t ∈ N and for every graph H , there exists η ∈ N and a nondecreas-ing function g such that the following hold:1. If G is a graph with no K s,t -subgraph and no H -minor, Y ⊆ V ( G ) with | Y | η , and L is an ( s, Y , , -list-assignment of G , then there exists an ( η, g ) -bounded L -coloring. . If G is a graph with no K s,t -subgraph and no odd H -minor, Y ⊆ V ( G ) with | Y | η , ℓ ∈ [ s + 2] and L is an ( s, Y , ℓ, s − -list-assignment of G , then there exists an ( η, g ) -bounded L -coloring such that for every x ∈ [ s + 3 , s + 1] , the set of vertices colored x is a stable set in G .Proof. Define the following: • Let f be the function f s,t in Lemma 3.5. • Let f : N → N be the identity function, and for every i ∈ N , let f i : N → N be thefunction defined by f i ( x ) := f i − ( x ) + f ( f i − ( x )) . • Let C be the integer c in Theorem 5.9. • Let t ′ := ⌈ ( C + 9) | V ( H ) | p log 12 | V ( H ) |⌉ . • Let θ be the number θ ∗ and let g , η be the functions g ∗ , η ∗ , respectively, in Lemma 5.8taking s = s , t = t and t ′ = t ′ . • Let ξ := 8 | V ( H ) | . • Let h : N → N be the function mentioned in Lemma 5.7 by taking s = s and t = t . • Let θ ′ := θ + t ′ + 1 . • Let η , g be the number and the function, respectively, mentioned in Lemma 5.1 bytaking s = s , t = t and k = 9 θ ′ . • Let η := h ( θ ′ + h (4 θ ′ ) + 1 + f ( h (4 θ ′ ) + 1)) . • Define η := η ( θ ′ ) + η + f ( η ) + h (4 θ ′ ) + 1 + f ( h (4 θ ′ ) + 1) + η . • Let η := h ( η + ξ ) + f ( h ( η + ξ )) . • Let g : N → N be the function defined by g ( x ) := x + g ( x ) + g ( x ) + η + η · η forevery x ∈ N .Suppose to the contrary that there exists a graph G with no K s,t -subgraph, a subset Y of V ( G ) with | Y | η , a number ℓ ∈ [ s + 2] , and list-assignment L of G such that the followinghold: • If G has no H -minor, then L is an ( s, Y , , -list-assignment such that there exists no ( η, g ) -bounded L -coloring of G . • If G has no odd H -minor, then L is an ( s, Y , ℓ, s − -list-assignment of G such thatthere exists no ( η, g ) -bounded L -coloring c of G such that { v ∈ V ( G ) : c ( v ) = x } is astable set in G for every x ∈ [ s + 3 , s + 1] .We further choose G and Y so that | V ( G ) | is as small as possible, and subject to this, | Y | isas large as possible. Let r := 0 and ℓ ′ := 0 when G has no H -minor; let r := s − and ℓ ′ := ℓ when G has no odd H -minor. Claim 5.10.1. Y = ∅ and N G ( Y ) = ∅ .Proof. If Y = ∅ , then let v be a vertex of G and define L ′ to be a ( { v } , { ℓ } ∪ [ s + 3 , s + 1]) -progress of L . Let Y ′ = { v } . By Lemma 4.4, L ′ is an ( s, Y ′ , , -list-assignment when G hasno H -minor; L ′ is an ( s, Y ′ , ℓ, s − -list-assignment when G has no odd H -minor. In addition, | Y ′ | η . So the maximality of Y implies that there exists an ( η, g ) -bounded L ′ -coloring c ′ of G such that if G has no odd H -minor, then { v ∈ V ( G ) : c ′ ( v ) = x } is a stable set in G forevery x ∈ [ s + 3 , s + 1] . But c ′ is an ( η, g ) -bounded L -coloring c of G such that if G has no120dd H -minor, then { v ∈ V ( G ) : c ( v ) = x } is a stable set in G for every x ∈ [ s + 3 , s + 1] , acontradiction.So Y = ∅ . Suppose that N G ( Y ) = ∅ . Let G ′ = G − Y . Then L | G ′ is an ( s, ∅ , , -list-assignment of G ′ when G has no H -minor; L | G ′ is an ( s, ∅ , ℓ, s − -list-assignment of G ′ when G has no odd H -minor. By the minimality of G , there exists an ( η, g ) -bounded L | G ′ -coloring c of G ′ such that if G has no odd H -minor, then { v ∈ V ( G ′ ) : c ( v ) = x } is a stable set in G ′ for every x ∈ [ s + 3 , s + 1] . By further coloring each vertex y in Y with the uniqueelement in L ( y ) , since | Y | | Y | g ( | Y | ) , there exists an ( η, g ) -bounded L -coloring of G suchthat if G has no odd H -minor, then { v ∈ V ( G ′ ) : c ( v ) = x } is a stable set in G ′ for every x ∈ [ s + 3 , s + 1] , a contradiction. This proves the claim. Claim 5.10.2. | Y | > θ ′ + 1 .Proof. Suppose | Y | θ ′ . So | Y | < η .Since η > η and g > g , if there exists an ( η , g ) -bounded L -coloring of G such thatfor every x ∈ [ s + 3 , s + 1] , the set of vertices colored x is a stable set in G , then it is an ( η, g ) -bounded L -coloring of G such that for every x ∈ [ s + 3 , s + 1] , the set of vertices colored x is a stable set in G , a contradiction. So there exists no ( η , g ) -bounded L -coloring of G such that for every x ∈ [ s + 3 , s + 1] , the set of vertices colored x is a stable set in G .Hence, by Lemma 5.1 (with taking F = [ s + 3 , s + 1] ) and Claim 5.10.1, there exist Y ′ ⊆ V ( G ) with | Y | < | Y ′ | η and an ( s, r +2 , Y ′ ) -list-assignment L ′ of G with L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) such that • there exists no ( η , g ) -bounded L ′ -coloring of G such that for every x ∈ [ s + 3 , s + 1] ,the set of vertices colored x is a stable set in G , • for every L ′ -coloring of G , every monochromatic component intersecting Y is containedin G [ Y ′ ] , • { y ∈ Y : L ( y ) ∩ [ s + 3 , s + 1] = ∅} = { y ∈ Y ′ : L ′ ( y ) ∩ [ s + 3 , s + 1] = ∅} , • for every x ∈ { ℓ } ∪ [ s + 3 , s + 1] and y ∈ Y ′ with x ∈ L ′ ( y ) , we have { v ∈ N G ( y ) − Y ′ : x ∈ L ′ ( v ) } = ∅ , and • for every v ∈ V ( G ) − Y ′ , we have L ′ ( v ) ∩ [ s + 3 , s + 1] = L ( v ) ∩ [ s + 3 , s + 1] .So L ′ is an ( s, Y ′ , , -list-assignment of G when G has no H -minor; L ′ is an ( s, Y ′ , ℓ, s − -list-assignment of G when G has no odd H -minor.Since η > | Y ′ | > | Y | , the maximality of Y implies that there exists an ( η, g ) -bounded L ′ -coloring c ′ of G such that for every x ∈ [ s + 3 , s + 1] , the set of vertices colored x is a stableset in G . So every c ′ -monochromatic component contains at most η g ( η ) vertices. Since c ′ is an L ′ -coloring, every c ′ -monochromatic component intersecting Y is contained in G [ Y ′ ] and hence contains at most | Y ′ | η | Y | g ( | Y | ) vertices. Since L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) , c ′ is an L -coloring. Therefore, c ′ is an ( η, g ) -bounded L -coloring of G such that forevery x ∈ [ s + 3 , s + 1] , the set of vertices colored x is a stable set in G , a contradiction.Define T = { ( A, B ) : | V ( A ∩ B ) | < θ ′ , | V ( A ) ∩ Y | θ ′ } to be a set of separations of G . Claim 5.10.3. T is a tangle in G of order θ ′ . roof. Suppose that T is not a tangle in G of order θ ′ . Since G has no K s,t -subgraph and L isan ( s, r + 2 , Y ) -list-assignment of G with η > | Y | > θ ′ + 1 by Claim 5.10.2. By Lemma 5.2by taking s = s, t = t, θ = θ ′ , η = η, g = g, r = r + 2 and F = { ℓ ′ } ∪ [ s + 3 , s + 1] , there existsan induced subgraph G ′ of G with | V ( G ′ ) | < | V ( G ) | , a subset Y ′ ⊆ V ( G ′ ) with | Y ′ | η , andan ( s, r + 2 , Y ′ ) -list-assignment L ′ of G ′ with L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) such that • there exists no ( η, g ) -bounded L ′ -coloring of G ′ such that for every x ∈ [ s + 3 , s + 1] ,the set of vertices colored x is a stable set, • { v ∈ Y ′ : ( { ℓ ′ } ∪ [ s + 3 , s + 1]) ∩ L ′ ( v ) = ∅} = { v ∈ Y : ( { ℓ ′ } ∪ [ s + 3 , s + 1]) ∩ L ( v ) = ∅} ∩ V ( G ′ ) , and • for every v ∈ V ( G ′ ) − Y ′ , we have L ′ ( v ) ∩ ( { ℓ ′ }∪ [ s +3 , s +1]) = L ( v ) ∩ ( { ℓ ′ }∪ [ s +3 , s +1]) .Hence L ′ is an ( s, Y , ℓ ′ , r ) -list-assignment of G ′ . This contradicts the minimality of G . Claim 5.10.4. T controls a K t ′ -minor.Proof. Suppose to the contrary that T does not control a K t ′ -minor. Note that θ ′ > θ , η > η ( θ ′ ) and g > g . By Lemma 5.8, since there does not exist an ( η, g ) -bounded L -coloringof G such that for every x ∈ [ s + 3 , s + 1] , the set of vertices color with x is a stable set in G ,we know there exist ( A ∗ , B ∗ ) ∈ T , a set Y A ∗ with | Y A ∗ | η ( θ ′ ) η and Y ∩ V ( A ∗ ) ⊆ Y A ∗ ⊆ V ( A ∗ ) , and an ( s, Y A ∗ , ℓ ′ , r ) -list-assignment L A ∗ of G [ V ( A ∗ )] such that there exists no ( η, g ) -bounded L A ∗ -coloring of G [ V ( A ∗ )] such that for every x ∈ [ s + 3 , s + 1] , the set of verticescolored x is a stable set in G [ V ( A ∗ )] . But | V ( A ∗ ) | < | V ( G ) | since | V ( A ∗ ) ∩ Y | < θ ′ < | Y | .So it contradicts the minimality of G .By Claim 5.10.4, G contains a K t ′ -minor α . In particular, G has no odd H -minor and L is an ( s, Y , ℓ, s − -list-assignment of G .By Theorem 5.9, either G contains an odd K | V ( H ) | -minor, or there exists a set Z ⊆ V ( G ) with | Z | < | V ( H ) | ξ such that the unique block U of G − Z intersecting all branchvertices of α disjoint from Z is bipartite. The former implies that G contains an odd H -minor, a contradiction. So the latter holds.Let ( A , B ) be the separation of G with V ( A ∩ B ) = Z and U ⊆ B , and subject tothis, | V ( A ) | is maximal, and subject to these, | E ( A ) | is minimal. Note that A − Z is theunion of all components of G − Z disjoint from U . For each cut-vertex v of G − Z containedin U , let ( A v , B v ) be the separation of G with V ( A v ∩ B v ) = Z ∪ { v } and A ∪ U ⊆ B v ,and subject to this, | V ( A v ) | is maximal, and subject to these, | E ( A v ) | is minimal. Define L = { ( A , B ) , ( A v , B v ) : v is a cut-vertex of G − Z contained in U } . Then L is a location bythe maximality of V ( A ) and V ( A v ) and the minimality of | E ( A ) | and | E ( A v ) | . Note thatfor every ( A, B ) ∈ L , the order of ( A, B ) is at most | Z | + 1 ξ < t ′ < θ ′ , so either ( A, B ) ∈ T or ( B, A ) ∈ T by Claim 5.10.3. Claim 5.10.5. L ⊆ T .Proof. Suppose that L 6⊆ T . Let ( A, B ) ∈ L − T . Then either ( A, B ) = ( A , B ) and ( B , A ) ∈ T , or there exists a cut-vertex v of G − Z contained in U such that ( A, B ) = ( A v , B v ) and ( B v , A v ) ∈ T . Since t ′ > | Z | + 2 , there exist vertices u , u of H such that Q , Q aredisjoint from Z , where Q , Q are the branch sets of α corresponding to u , u , respectively.By the definition of U , U intersects Q ∪ Q . Since Q ∩ Q = ∅ and U ⊆ V ( B ) , one of Q , Q B − V ( A ) . Since T controls α and ( B, A ) ∈ T has order less than t ′ , both Q , Q intersects V ( A ) − V ( B ) , a contradiction. Hence L ⊆ T .Let L ′ be a ( Z, ℓ ) -growth of L , and let Y ′ = { v ∈ V ( G ) : | L ′ ( v ) | = 1 } . By Lemma 5.7, | Y ′ | h ( | Y ∪ Z | ) h ( η + ξ ) and for every ( A, B ) ∈ L , we have | Y ′ ∩ V ( A ) | h ( | A ∩ B | + | Y ∩ V ( A ) | ) h (4 θ ′ ) . Claim 5.10.6. For every v ∈ V ( G ) − Y ′ , we have L ′ ( v ) ∩ ( { ℓ } ∪ [ s + 3 , s + 1]) = ∅ and L ′ ( v ) − ( { ℓ } ∪ [ s + 3 , s + 1]) = ∅ .Proof. By Lemma 5.7, L ′ is an ( s, Y ′ , ℓ, s − -list-assignment of G . So L ′ is an ( s, s + 1 , Y ′ ) -list-assignment of G . By (L5), | L ′ ( v ) | > s + 2 for every v ∈ V ( G ) − Y ′ . Since |{ ℓ } ∪ [ s +3 , s + 1] | = s , we have L ′ ( v ) − ( { ℓ } ∪ [ s + 3 , s + 1]) = ∅ for every v ∈ V ( G ) − Y ′ . Since | [2 s + 1] − ( { ℓ } ∪ [ s + 3 , s + 1]) | = s + 1 , we have L ′ ( v ) ∩ ( { ℓ } ∪ [ s + 3 , s + 1]) = ∅ for every v ∈ V ( G ) − Y ′ .Let G ′ = G [( T ( A,B ) ∈L V ( B )) ∪ Y ′ ] . Note that G ′ = G [ V ( U ) ∪ Y ′ ] . Observe that G ′ − Y ′ ⊆ U ,so G ′ − Y ′ is bipartite. Let { P, Q } be a bipartition of G ′ − Y ′ . Define an L ′ | G ′ -coloring c ′ of G ′ as follows. • If v ∈ Y ′ , then let c ′ ( v ) be the unique element of L ′ ( v ) . • If v ∈ P , then let c ′ ( v ) be an element of L ′ ( v ) ∩ ( { ℓ } ∪ [ s + 3 , s + 1]) . • If v ∈ Q , then let c ′ ( v ) be an element of L ′ ( v ) − ( { ℓ } ∪ [ s + 3 , s + 1]) .Note that c ′ is well-defined by Claim 5.10.6. Claim 5.10.7. Every c ′ -monochromatic component contains at most η vertices. Further-more, for every x ∈ [ s + 3 , s + 1] , the set of vertices colored x is a stable set in G ′ .Proof. Let x ∈ [2 s + 1] , and let M be a c ′ -monochromatic component such that all verticesof M are colored x . If V ( M ) ∩ Y ′ = ∅ , then V ( M ) ⊆ P or V ( M ) ⊆ Q , so M consists of onevertex as P and Q are stable sets in G ′ . So we may assume that V ( M ) ∩ Y ′ = ∅ .Since L ′ is an ( s, Y ′ , ℓ, s − -list-assignment of G , by (R3), if x ∈ { ℓ } ∪ [ s + 3 , s + 1] , theneither V ( M ) ∩ Y ′ = ∅ or V ( M ) ⊆ Y ′ . Recall V ( M ) ∩ Y ′ = ∅ , so if x ∈ { ℓ } ∪ [ s + 3 , s + 1] ,then V ( M ) ⊆ Y ′ . Hence, if x ∈ [ s + 3 , s + 1] , then V ( M ) ⊆ Y ′ , so M consists of one vertexby (R4); if x = ℓ , then V ( M ) ⊆ Y ′ , so M contains at most | Y ′ | h ( η + ξ ) η vertices.So we may assume that x ∈ [ s + 2] − { ℓ } . In particular, V ( M ) ∩ P = ∅ . Since L ′ isan ( s, Y ′ , ℓ, s − -list-assignment, L ′ is an ( s, s + 1 , Y ′ ) -list-assignment, so | N G ( v ) ∩ Y ′ | > s for every vertex v ∈ V ( M ) − Y ′ with N G ( v ) ∩ Y ′ ∩ V ( M ) = ∅ by (L3). By Lemma 3.5, |{ v ∈ V ( M ) − Y ′ : N G ( v ) ∩ Y ′ ∩ V ( M ) = ∅}| f ( | Y ′ | ) . Note that every component of M − Y ′ is contained in G ′ [ Q ] which is a graph with no edge. Since V ( M ) ∩ Y ′ = ∅ , every vertex in V ( M ) − Y ′ has a neighbor in Y ′ ∩ V ( M ) . So { v ∈ V ( M ) − Y ′ : N G ( v ) ∩ Y ′ ∩ V ( M ) = ∅} = V ( M ) − Y ′ . Hence | V ( M ) | | V ( M ) ∩ Y ′ | + | V ( M ) − Y ′ | | Y ′ | + f ( | Y ′ | ) h ( η + ξ ) + f ( h ( η + ξ )) = η .Let Y ′′ := V ( G ′ ) . Note that Y ′′ ⊇ Y ′ . Let L ′′ be the following list-assignment of G : • For every v ∈ Y ′ , let L ′′ ( v ) := L ′ ( v ) . • For every v ∈ V ( G ′ ) − Y ′ , let L ′′ ( v ) := { c ′ ( v ) } .123 For every v ∈ N The following hold: • L ′′ is an ( s, s + 1 , Y ′′ ) -list-assignment. • For every x ∈ [ s + 3 , s + 1] , { y ∈ Y ′′ : x ∈ L ′′ ( y ) } = { y ∈ Y ′ : x ∈ L ′ ( y ) } ∪ { y ∈ V ( G ′ ) − Y ′ : c ′ ( y ) = x } is a stable set in G . • For every v ∈ N For every s ∈ N , there exists λ ∈ N such that every odd K s +1 -minor-freegraph has an (8 s − -coloring with clustering λ .Proof. By Theorem 1.16, there exists λ such that every odd K s +1 -minor-free graph with no K s − , ( s − ) + s +2 -subgraph has a (2(2 s − 1) + 1) -coloring with clustering λ . So every odd K s +1 -minor-free graph with no bipartite K ∗ s,s +1 -subdivision has a (4 s − -coloring with clustering λ . We may assume that λ > s − . The theorem now follows from Lemma 5.11. This section proves our results for graphs excluding a subdivision (Theorems 1.4 and 1.6to 1.8). Section 6.1 introduces a structure theorem of the first author and Thomas [47] forgraphs excluding a fixed subdivision, and uses it to prove Theorem 1.7. Building on thiswork, Section 6.2 proves the remaining theorems about subdivisions.126he proofs in this section rely on the notion of ( W, F ) -progress introduced in Section 4.2.However, we only need the F = ∅ case. So we define a W -progress to be a ( W, ∅ ) -progress.The following two results are simply Lemmas 4.4 and 5.1 with F = ∅ . Lemma 6.1. Let s, r ∈ N and L be an ( s, r, Y ) -list-assignment of a graph G . Let W ⊆ V ( G ) .Then every W -progress L ′ of L is an ( s, r, Y ∪ W ) -list-assignment of G , and L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) . Lemma 6.2. For all s, t, k ∈ N , there exist a number η > k and a nondecreasing function g with domain N and with g (0) > η such that if G is a graph with no K s,t -subgraph, r ∈ N , Y is a subset of V ( G ) with | Y | η , and L is an ( s, r, Y ) -list-assignment of G , then at leastone of the following holds:1. There exists an ( η, g ) -bounded L -coloring of G .2. | Y | > k .3. For every color ℓ , there exist a subset Y ′ of V ( G ) with η > | Y ′ | > | Y | and an ( s, r, Y ′ ) -list-assignment L ′ of G with L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) , such that:(a) there does not exist an ( η, g ) -bounded L ′ -coloring c ′ of G ,(b) for every L ′ -coloring of G , every monochromatic component intersecting Y is con-tained in G [ Y ′ ] , and(c) for every y ∈ Y ′ with ℓ ∈ L ′ ( y ) , we have { v ∈ N G ( y ) − Y ′ : ℓ ∈ L ′ ( v ) } = ∅ .4. Y = ∅ , N G ( Y ) = ∅ , and there does not exist an ( η, g ) -bounded L | G − Y -coloring of G − Y . We us the following two previous results about separations and tangles. The first is simplyLemma 5.2 with F = ∅ . Lemma 6.3. For all s, t, θ, η, r ∈ N with η > θ + 1 , for every nondecreasing function g withdomain N , if G is a graph with no K s,t -subgraph, Y is a subset of V ( G ) with θ +1 | Y | η ,and L is an ( s, r, Y ) -list-assignment of G , then at least one of the following holds:1. There exists an ( η, g ) -bounded L -coloring of G .2. There exist an induced subgraph G ′ of G with | V ( G ′ ) | < | V ( G ) | , a subset Y ′ of V ( G ′ ) with | Y ′ | η and an ( s, r, Y ′ ) -list-assignment L ′ of G ′ such that:(a) L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ′ ) .(b) There does not exist an ( η, g ) -bounded L ′ -coloring of G ′ .3. T := { ( A, B ) : | V ( A ∩ B ) | < θ, | V ( A ) ∩ Y | θ } is a tangle of order θ in G . The next lemma is Lemma 5.8 with ℓ = r = 0 . Lemma 6.4. For all s, t, t ′ ∈ N , there exist θ ∗ ∈ N and nondecreasing functions g ∗ , η ∗ withdomain N such that if G is a graph with no K s,t -subgraph, θ ∈ N with θ > θ ∗ , η ∈ N with η > η ∗ ( θ ) , Y ⊆ V ( G ) with θ < | Y | η , L is a restricted ( s, , Y ) -list-assignment of G , g is a nondecreasing function with domain N with g > g ∗ , and T := { ( A, B ) : | V ( A ∩ B ) | <θ, | V ( A ) ∩ Y | θ } is a tangle in G of order θ that does not control a K t ′ -minor, then either: . there exists an ( η, g ) -bounded L -coloring of G , or2. there exist ( A ∗ , B ∗ ) ∈ T , a set Y A ∗ with | Y A ∗ | η ∗ ( θ ) and Y ∩ V ( A ∗ ) ⊆ Y A ∗ ⊆ V ( A ∗ ) ,and a restricted ( s, , Y A ∗ ) -list-assignment L A ∗ of G [ V ( A ∗ )] such that there exists no ( η, g ) -bounded L A ∗ -coloring of G [ V ( A ∗ )] . The following theorem is a special case of a theorem by the first author and Thomas [47] Theorem 6.5 ([47, Theorem 6.8]) . For any integers d, h and graph H on h vertices withmaximum degree at most d , there exist integers θ, ξ such that if G is a graph containing no H -subdivision, and if T is a tangle in G of order at least θ controlling a K ⌊ dh ⌋ -minor, thenthere exists Z ⊆ V ( G ) with | Z | ξ such that for every vertex v ∈ V ( G ) − Z , there exists ( A, B ) ∈ T − Z of order less than d such that v ∈ V ( A ) − V ( B ) . The next two lemmas imply Theorem 1.7, since if s, d ∈ N and d + s < , then d = 1 . Lemma 6.6. If H is a graph of maximum degree at most 1, then every graph with no H -subdivision is 2-colorable with clustering max { | V ( H ) | − , } .Proof. Since H is of maximum degree at most one, G has no H -subdivision implies that G does not contain a matching of size | V ( H ) | , and hence G contains a vertex-cover S of size atmost | V ( H ) | − . By coloring every vertex in S with 1 and coloring every vertex in V ( G ) − S with 2, we obtain a 2-coloring of G with clustering max {| S | , } max { | V ( H ) | − , } . Lemma 6.7. For any s, t, d ∈ N and graph H of maximum degree d with d + s > , thereexists η ∈ N and a nondecreasing function g such that if G is a graph with no K s,t -subgraphand no H -subdivision, Y ⊆ V ( G ) with | Y | η and L is a restricted ( s ′ , , Y ) -list-assignmentof G , then there exists an ( η, g ) -bounded L -coloring, where s ′ = 3 d + s − .Proof. Define the following: • Let f be the function f s,t mentioned in Lemma 3.5. • Let θ be the number θ ∗ and g , η be the functions g ∗ , η ∗ , respectively, mentioned inLemma 6.4 by taking s = s ′ , t = t and t ′ = ⌊ d | V ( H ) |⌋ . • Let θ and ξ be the numbers θ and ξ mentioned in Theorem 6.5, respectively, by taking d = d , h = | V ( H ) | and H = H . • Let a := f ( ξ ) d + ξ + 1 , and let a i := da i − + 1 for i ∈ N . • Let θ := θ + θ + ( d − a ( d − a . • Let η be the number η and let g be the function g mentioned in Lemma 6.2 by taking s = s ′ , t = t and k = 9 θ . Note that g (0) > η > θ by Lemma 6.2. • Let η := η ( θ ) + η + ( d − a ( d − a . • Let g : N → N be the function defined by g (0) := g (0) + g (0) and g ( x + 1) := g ( x + 1) + g ( x + 1) + P xi =0 i g ( i ) for x ∈ N .Let G be a graph with no K s,t -subgraph and with no subdivision of H , let Y ⊆ V ( G ) with | Y | η , and let L be a restricted ( s ′ , , Y ) -list-assignment of G . Suppose to the contrary thatthere exists no ( η, g ) -bounded L -coloring of G . We further assume that | V ( G ) | is minimum,and subject to this, | Y | is maximum. 128 laim 6.7.1. Y = ∅ and N G ( Y ) = ∅ .Proof. First suppose that Y = ∅ . Let v be a vertex of G , and let L ′ be a { v } -progress of L .Let Y ′ = { v } . By Lemma 6.1, L ′ is an ( s ′ , , Y ′ ) -list-assignment of G . Since | Y ′ | η , themaximality of | Y | implies that there exists an ( η, g ) -bounded L ′ -coloring c ′ of G . But c ′ is an ( η, g ) -bounded L -coloring c of G , a contradiction.So Y = ∅ . Suppose that N G ( Y ) = ∅ . Let G ′ := G − Y . Then L | G ′ is an ( s ′ , , ∅ ) -list-assignment of G − Y . By the minimality of | V ( G ) | , there exists an ( η, g ) -bounded L | G ′ -coloring c of G ′ . Color each vertex y in Y with the unique element in L ( y ) . Since | Y | | Y | g ( | Y | ) ,we obtain an ( η, g ) -bounded L -coloring of G , a contradiction. Claim 6.7.2. | Y | > θ + 1 .Proof. Suppose | Y | θ . So | Y | < η . Since G has no K s,t -subgraph, G has no K s ′ ,t -subgraph. Applying Lemma 6.2 and Claim 6.7.1, either there exists an ( η , g ) -bounded L -coloring of G , or there exist Y ′ ⊆ V ( G ) with η > | Y ′ | > | Y | and an ( s ′ , , Y ′ ) -list-assignment L ′ of G with L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) such that for every L ′ -coloringof G , every monochromatic component intersecting Y is contained in G [ Y ′ ] . Since η η and g g , every ( η , g ) -bounded L -coloring of G is an ( η, g ) -bounded L -coloring of G ,so the former does not hold. Hence there exist Y ′ ⊆ V ( G ) with η > | Y ′ | > | Y | and arestricted ( s ′ , , Y ′ ) -list-assignment L ′ of G with L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) such thatfor every L ′ -coloring of G , every monochromatic component intersecting Y is contained in G [ Y ′ ] . Since | Y ′ | η η , the maximality of | Y | implies that there exists an ( η, g ) -bounded L ′ -coloring c ′ of G . So every monochromatic component respect to c ′ contains at most η g ( η ) vertices. Since L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) , c ′ is also an L -coloring of G . Every c ′ -monochromatic component intersecting Y is contained in G [ Y ′ ] and hence contains at most | Y ′ | η g (0) g (0) | Y | g ( | Y | ) vertices. So c ′ is an ( η, g ) -bounded L -coloring of G , acontradiction.Let T be the set of separations ( A, B ) of G such that | V ( A ∩ B ) | < θ and | V ( A ) ∩ Y | θ . Claim 6.7.3. T is a tangle in G of order θ .Proof. Suppose that T is not a tangle in G of order θ . Note that G has no K s ′ ,t -subgraphand L is an ( s ′ , , Y ) -list-assignment of G with η > | Y | > θ + 1 by Claim 6.7.2. ApplyingLemma 6.3 by taking s = s ′ , t = t , θ = θ , η = η , r = 2 and g = g , there exists aninduced subgraph G ′ of G with | V ( G ′ ) | < | V ( G ) | , a subset Y ′ ⊆ V ( G ′ ) with | Y ′ | η , andan ( s ′ , , Y ′ ) -list-assignment L ′ of G ′ with L ′ ( v ) ⊆ L ( v ) for every v ∈ V ( G ) such that thereexists no ( η, g ) -bounded L ′ -coloring of G ′ . This contradicts the minimality of | V ( G ) | . Claim 6.7.4. T controls a K ⌊ d | V ( H ) |⌋ -minor.Proof. Suppose to the contrary that T does not control a K ⌊ d | V ( H ) |⌋ -minor. Note that θ > θ , η > η ( θ ) and g > g . Apply Lemma 6.4 with s = s ′ , t = t and t ′ = ⌊ d | V ( H ) |⌋ . Since theredoes not exist an ( η, g ) -bounded L -coloring of G , we know there exist ( A ∗ , B ∗ ) ∈ T , a set Y A ∗ with | Y A ∗ | η ( θ ) η and Y ∩ V ( A ∗ ) ⊆ Y A ∗ ⊆ V ( A ∗ ) , and a restricted ( s ′ , , Y A ∗ ) -list-assignment L A ∗ of G [ V ( A ∗ )] such that there exists no ( η, g ) -bounded L A ∗ -coloring of G [ V ( A ∗ )] .But | V ( A ∗ ) | < | V ( G ) | since | V ( A ∗ ) ∩ Y | < θ < | Y | . This contradicts the minimality of | V ( G ) | . 129ince G contains no subdivision of H , by Theorem 6.5 and Claim 6.7.4, there exists Z ⊆ V ( G ) with | Z | ξ such that for every v ∈ V ( G ) − Z , there exists ( A v , B v ) ∈ T − Z oforder at most d − such that v ∈ V ( A v ) − V ( B v ) .We may assume that for every v ∈ V ( G ) − Z ,(i) ( A v , B v ) ∈ T − Z has order at most d − and v ∈ V ( A v ) − V ( B v ) ,(ii) subject to (i), A v − V ( A v ∩ B v ) is connected,(iii) subject to (i) and (ii), every vertex in V ( A v ∩ B v ) is adjacent to some vertex in V ( A v ) − V ( B v ) ,(iv) subject to (i)–(iii), V ( A v ) is maximal,(v) subject to (i)–(iv), | V ( A v ∩ B v ) | is minimal, and(vi) subject to (i)–(v), A v is maximal.Note that for every v ∈ V ( G ) − Z , A v is connected and for every two vertices x, y ∈ V ( A v ) ,there exists a path in A v from x to y internally disjoint from V ( A v ∩ B v ) since A v − V ( A v ∩ B v ) is connected and every vertex in V ( A v ∩ B v ) is adjacent to some vertex in V ( A v ) − V ( B v ) For any subset C ⊆ T − Z , let ( A C , B C ) be the separation ( S ( A,B ) ∈C A, T ( A,B ) ∈C B ) . Notethat V ( A C ∩ B C ) ⊆ S ( A,B ) ∈C V ( A ∩ B ) , so | V ( A C ∩ B C ) | |C| ( d − . Claim 6.7.5. Let C = { ( A w , B w ) : w ∈ W } for some W ⊆ V ( G ) − Z . If x is a vertex in V ( A C ∩ B C ) , then V ( A x ∩ B x ) − V ( B w ) = ∅ for some w ∈ V ( G ) − Z with ( A w , B w ) ∈ C .Proof. Since x ∈ V ( A C ∩ B C ) , there exists w ∈ W ⊆ V ( G ) − Z such that ( A w , B w ) ∈ C and x ∈ V ( A w ∩ B w ) . Suppose to the contrary that V ( A x ∩ B x ) ⊆ V ( B w ) .First suppose that there exists v ∈ V ( A w ) − ( V ( B w ) ∪ V ( A x )) . Since A w − V ( B w ) isconnected by (ii) and every vertex in V ( A w ∩ B w ) is adjacent to a vertex in V ( A w ) − V ( B w ) by(iii), there exists a path P in G [( V ( A w ) − V ( B w )) ∪{ x } ] from x to v . Since x ∈ V ( A x ) − V ( B x ) and v ∈ V ( G ) − ( Z ∪ V ( A x )) , P − x intersects V ( A x ∩ B x ) ⊆ V ( B w ) . But V ( P − x ) ⊆ V ( A w ) − V ( B w ) , a contradiction. So V ( A w ) − V ( B w ) ⊆ V ( A x ) .Suppose that there exists a vertex u ∈ V ( A w ∩ B w ) − V ( A x ) . Since u ∈ V ( A w ∩ B w ) ,there exists u ′ ∈ N G ( u ) ∩ V ( A w ) − V ( B w ) by (iii). So u ′ ∈ N G ( u ) ∩ V ( A w ) − V ( B w ) ⊆ N G ( u ) ∩ V ( A x ) . Since u V ( A x ) , u ′ ∈ V ( A x ∩ B x ) ∩ V ( A w ) − V ( B w ) , contradicting theassumption V ( A x ∩ B x ) ⊆ V ( B w ) . Hence V ( A w ∩ B w ) ⊆ V ( A x ) .Therefore, V ( A w ) ⊆ V ( A x ) . By (v), every vertex in V ( A x ∩ B x ) is adjacent to some vertexin V ( B x ) − V ( A x ) . So if V ( A x ) = V ( A w ) , then V ( A x ∩ B x ) ⊆ V ( A w ∩ B w ) , and since ( A w , B w ) satisfies (v), V ( B w ) = V ( B x ) . Hence if V ( A x ) = V ( A w ) , then ( A x , B x ) = ( A w , B w ) by (vi).Since x ∈ V ( A x ) − V ( B x ) and x ∈ V ( A w ∩ B w ) , ( A x , B x ) = ( A w , B w ) . So V ( A w ) ⊂ V ( A x ) .Since ( A w , B w ) satisfies (iv), w ∈ V ( B x ) . Since V ( A x ∩ B x ) ⊆ V ( B w ) and w V ( B w ) , w ∈ V ( B x ) − V ( A x ) . So V ( A w ) V ( A x ) , a contradiction.Let Z ′ := { v ∈ V ( G ) − ( Y ∪ Z ) : | N G ( v ) ∩ Z | > s } . Note that | Z ′ | f ( | Z | ) f ( ξ ) byLemma 3.5.We say that a triple ( C , S, T ) is useful if the following hold:(U1) There exists W ⊆ V ( G ) − Z such that C = { ( A v , B v ) : v ∈ W } .(U2) N G [ N G [ Z ′ ]] ∩ V ( B C ) = ∅ . 130U3) S is a subset of N G [ V ( A C ∩ B C )] ∩ V ( A C ) and T is a subset of Y ∩ V ( A C ) − V ( B C ) suchthat there exists a bijection ι from a subset of Y ∩ V ( A C ) to S such that: – | S | + | T | + | Z | + 1 | Y ∩ V ( A C ) − V ( B C ) | + |{ y ∈ Y ∩ S ∩ V ( A C ∩ B C ) : ι ( y ) = y }| ,and – for every vertex y in the domain of ι , ∗ if y ∈ V ( A C ) − V ( B C ) and there exists a vertex v ∈ N G ( y ) ∩ V ( A C ∩ B C ) − S ,then ι ( y ) ∈ N G ( y ) ∩ V ( A C ∩ B C ) , and ∗ if y ∈ V ( A C ∩ B C ) , then ι ( y ) = y .(U4) T is disjoint from Z ′ and the domain of ι . Claim 6.7.6. There exists a collection C of members of T − Z with |C| | Z ′ | d + | Z | + 1 such that ( C , ∅ , ∅ ) is useful.Proof. For every u ∈ V ( G ) − Z , let C u := { ( A u , B u ) , ( A v , B v ) : v ∈ N G ( u ) ∩ V ( B u ) } . Notethat | N G ( u ) ∩ V ( B u ) | | V ( A u ∩ B u ) | d − since u ∈ V ( A u ) − V ( B u ) . So |C u | d . Notethat N G [ { u } ] ∩ V ( B C u ) = ∅ .For every u ∈ V ( G ) − Z , let C ′ u := C u ∪ { ( A v , B v ) : v ∈ N G ( N G [ { u } ]) ∩ V ( B C u ) } . Note that | N G ( N G [ { u } ]) ∩ V ( B C u ) | | V ( A C u ∩ B C u ) | ( d − |C u | ( d − d . So |C ′ u | |C u | + ( d − d d . Note that N G [ N G [ { u } ]] ∩ V ( B C ′ u ) = ∅ .Let C ′ := S z ∈ Z ′ C ′ z . Then N G [ N G [ Z ′ ]] ∩ V ( B C ′ ) = ∅ . And |C ′ | | Z ′ | d . Since | Y − Z | > | Y | − | Z | > θ − ξ > θ > | Z | , there exists a subset Y of Y − Z with | Y | = | Z | + 1 . Let C := C ′ ∪ { ( A y , B y ) : y ∈ Y } . Clearly ( C , ∅ , ∅ ) satisfies (U1) and (U4). Since B C ′ ⊇ B C , ( C , ∅ , ∅ ) satisfies (U2). Since Y ⊆ V ( A C ) − V ( B C ) , | Z | + 1 = | Y | | Y ∩ V ( A C ) − V ( B C ) | , so ( C , ∅ , ∅ ) satisfies (U3). Note that | C | |C ′ | + | Y | | Z ′ | d + | Z | + 1 .For a useful triple ( C , S, T ) , a vertex v of V ( G ) − Z is: • ( C , S, T ) -dangerous if v ∈ V ( A C ∩ B C ) − S and there exists v ′ ∈ N G ( v ) ∩ V ( A C ) − ( V ( B C ) ∪ S ) such that either: – v ′ Y and | (( Y ∩ V ( A C )) ∪ ( S − V ( A C ∩ B C ))) ∩ N G ( v ′ ) | > d − , or – v ′ ∈ Y − T , • ( C , S, T ) -heavy if v ∈ V ( A C ∩ B C ) − S and | N G ( v ) ∩ (( Y ∩ V ( A C ) − V ( B C )) ∪ ( S − V ( A C ∩ B C ))) | > d − . Claim 6.7.7. Let ( C , S, T ) be a useful triple and let x ∈ V ( A C ∩ B C ) be a ( C , S, T ) -heavyvertex. Then there exists a useful triple ( C ′ , S ′ , T ′ ) with C ′ = C ∪ { ( A x , B x ) } , such that: • V ( A C ′ ∩ B C ′ ) − S ′ ⊆ V ( A C ∩ B C ) − S , • the set of ( C ′ , S ′ , T ′ ) -heavy vertices is strictly contained in the set of ( C , S, T ) -heavyvertices, and • the set of ( C ′ , S ′ , T ′ ) -dangerous vertices is a subset of the set of ( C , S, T ) -dangerousvertices. roof. Let C ′ := C ∪ { ( A x , B x ) } . Let X := N G ( x ) ∩ ( Y ∪ S ) ∩ V ( A x ∩ A C ) − V ( B x ∪ B C ) . Since x ∈ V ( A x ) − V ( B x ) , N G ( x ) ⊆ V ( A x ) . So | X | > | N G ( x ) ∩ ( Y ∪ S ) ∩ V ( A C ) − V ( B C ) | − | V ( A x ∩ B x ∩ A C ) − V ( B C ) | > d − − | V ( A x ∩ B x ∩ A C ) − V ( B C ) | since x is ( C , S, T ) -heavy. That is, | V ( A x ∩ B x ) − V ( B C ) | = | V ( A x ∩ B x ∩ A C ) − V ( B C ) | > d − − | X | . Since x is ( C , S, T ) -heavy, x S . Let ι be a bijection mentioned in (U3) witnessing that ( C , S, T ) is useful. Let X ′ bethe intersection of X and the domain of ι .For each y ∈ X ′ , since x ∈ N G ( y ) ∩ V ( A C ∩ B C ) − S , ι ( y ) ∈ N G ( y ) ∩ V ( A C ∩ B C ) by (U3). Since X ′ ⊆ X ⊆ V ( A x ∩ A C ) − V ( B x ∪ B C ) , for each y ∈ X ′ , N G ( y ) ⊆ V ( A x ) , so ι ( y ) ∈ N G ( y ) ∩ V ( A C ∩ B C ) ⊆ V ( A x ) ∩ V ( A C ∩ B C ) ⊆ ( V ( A C ′ ) − V ( B C ′ )) ∪ V ( A C ∩ B C ∩ A x ∩ B x ) .Let Z := V ( A x ∩ B x ) − V ( A C ) Z := V ( A x ∩ B x ∩ A C ∩ B C ) − { ι ( y ) : y ∈ X ′ } Z := V ( A x ∩ B x ∩ A C ∩ B C ) ∩ { ι ( y ) : y ∈ X ′ } . So { Z , Z , Z } is a partition of V ( A x ∩ B x ∩ B C ) , and hence | Z ∪ Z ∪ Z | = | V ( A x ∩ B x ∩ B C ) | = | V ( A x ∩ B x ) | − | V ( A x ∩ B x ) − V ( B C ) | d − − | V ( A x ∩ B x ) − V ( B C ) | . Recall that | V ( A x ∩ B x ) − V ( B C ) | > d − −| X | . So | Z ∪ Z ∪ Z | ( d − − ( d − −| X | ) = | X | .Let S ′ := (cid:0) S ∩ V ( B x ) − V ( A x ∩ B x ∩ A C ∩ B C ) (cid:1) ∪ (cid:0) V ( A C ′ ∩ B C ′ ) − V ( A C ∩ B C ) (cid:1) ∪ V ( A x ∩ B x ∩ A C ∩ B C ) . Note that ( V ( A C ′ ∩ B C ′ ) − V ( A C ∩ B C ) ∪ V ( A x ∩ B x ∩ A C ∩ B C ) ⊆ V ( A x ∩ B x ∩ B C ) = Z ∪ Z ∪ Z .So | S ′ | ( | S ∩ V ( B x ) | − | S ∩ V ( A x ∩ B x ∩ A C ∩ B C ) | )+ | Z ∪ Z ∪ Z | | S ∩ V ( B x ) | − |{ y ∈ X ′ : ι ( y ) ∈ Z }| + | X | = | S ∩ V ( B x ) | + | X − { y ∈ X ′ : ι ( y ) ∈ Z }| = |{ y ∈ Y ∩ V ( A C ) : ι ( y ) ∈ S ∩ V ( B x ) }| + | X − X ′ | + | X ′ − { y ∈ X ′ : ι ( y ) ∈ Z }| . Recall that for every y ∈ X ′ , ι ( y ) ∈ N G ( y ) ∩ V ( A C ∩ B C ) ∩ V ( A x ) . So if y ∈ X ′ − { y ∈ X ′ : ι ( y ) ∈ Z } , then ι ( y ) ∈ N G ( y ) ∩ V ( A C ∩ B C ) ∩ V ( A x ) − V ( A x ∩ B x ∩ A C ∩ B C ) = N G ( y ) ∩ V ( A C ∩ B C ∩ A x ) − V ( B x ) , so ι ( y ) S ∩ V ( B x ) . That is, { y ∈ Y ∩ V ( A C ) : ι ( y ) ∈ S ∩ V ( B x ) } and X ′ −{ y ∈ X ′ : ι ( y ) ∈ Z } are disjoint. Note that X − X ′ is disjoint from the domain of ι . So { y ∈ Y ∩ V ( A C ) : ι ( y ) ∈ S ∩ V ( B x ) } , X − X ′ and X ′ − { y ∈ X ′ : ι ( y ) ∈ Z } are pairwise disjoint sets. Therefore, | S ′ | |{ y ∈ Y ∩ V ( A C ) : ι ( y ) ∈ S ∩ V ( B x ) } ∪ ( X − { y ∈ X ′ : ι ( y ) ∈ Z } ) | . Since X ⊆ Y ∪ S and X ∩ V ( B x ) = ∅ , for every x ∈ X − { y ∈ X ′ : ι ( y ) ∈ Z } ,if x X ∩ Y − { y ∈ X ′ : ι ( y ) ∈ Z } , then x ∈ X ∩ S − Y and ι ( y ) = x for some132 ∈ Y ∩ V ( A C ) such that ι ( y ) S ∩ V ( B x ) . In addition, if y is a vertex in Y ∩ V ( A C ) suchthat ι ( y ) ∈ X ∩ S − Y , then ι ( y ) S ∩ V ( B x ) .Since | S ′ | |{ y ∈ Y ∩ V ( A C ) : ι ( y ) ∈ S ∩ V ( B x ) } ∪ ( X − { y ∈ X ′ : ι ( y ) ∈ Z } ) | , thereexists an injection ι ′ such that • ι ′ ( y ) = ι ( y ) if y is in the domain of ι and ι ( y ) ∈ S ∩ V ( B x ) , • for each v ∈ S ′ − ( S ∩ V ( B x )) , there exists exactly one element y ∈ ( X ∩ Y − { y ∈ X ′ : ι ( y ) ∈ Z } ) ∪ { y ∈ Y ∩ V ( A C ) : ι ( y ) ∈ X ∩ S − Y } such that ι ′ ( y ) = v , and • if ι ( y ) = ι ′ ( y ) for some y , y , then y = y .Recall that ι ( y ) S ∩ V ( B x ) for every y ∈ ( X ∩ Y − { y ∈ X ′ : ι ( y ) ∈ Z } ) ∪ { y ∈ Y : ι ( y ) ∈ X ∩ S − Y } . Then ι ′ is a bijection from a subset of Y ∩ V ( A C ′ ) to S ′ . We further modify ι ′ and S ′ by applying the following operations for some vertex y ∈ V ( A C ′ ) − V ( B C ′ ) in thedomain of ι ′ with ι ′ ( y ) N G ( y ) ∩ V ( A C ′ ∩ B C ′ ) and N G ( y ) ∩ V ( A C ′ ∩ B C ′ ) − S ′ = ∅ , and thenrepeating until no such vertex y exists: • add a vertex v ∈ N G ( y ) ∩ V ( A C ′ ∩ B C ′ ) − S ′ into S ′ , • delete ι ′ ( y ) from S ′ , and • redefine ι ′ ( y ) to be v .”Now, further modify ι ′ and S ′ by applying the following operations for some vertex z ∈ S ′ − N G [ V ( A C ′ ∩ B C ′ )] , and repeating until no such vertex z exists: • remove z from S ′ , and • if y is the element in the domain of ι ′ with ι ′ ( y ) = z , then remove y from the domain of ι ′ .Notice that for each vertex z removed from S ′ in the above procedure, z ∈ S − V ( A C ∩ B C ) and N G ( z ) ∩ V ( A C ∩ B C ) ⊆ V ( A C ∩ B C ) − V ( B x ) . Note that ι ′ remains a bijection from asubset of Y ∩ V ( A C ′ ) to S ′ .Observe that for every y in the domain of ι ′ with y ∈ V ( A ′C ) − V ( B ′C ) and N G ( y ) ∩ V ( A C ′ ∩ B C ′ ) − S ′ = ∅ , ι ′ ( y ) ∈ N G ( y ) ∩ V ( A C ′ ∩ B C ′ ) due to the above modification. In addition, if y is in the domain of ι ′ and y ∈ V ( A C ′ ∩ B C ′ ) , then y ∈ V ( A C ∩ B C ) and y is in the domain of ι such that ι ( y ) = ι ′ ( y ) , so ι ′ ( y ) = ι ( y ) = y .Let T ′ be the set obtained from T by deleting the domain of ι ′ . So T ′ is disjoint fromthe domain of ι ′ . Since T is disjoint from Z ′ , T ′ is disjoint from Z ′ . So ( C ′ , S ′ , T ′ ) satisfies(U4). In addition, | S ′ | − | S | is at most the number of vertices in X and in the domain of ι ′ but not in the domain of ι . So | S ′ | − | S | | T | − | T ′ | . Hence | S ′ | + | T ′ | | S | + | T | .Since { y ∈ Y ∩ S ∩ V ( A C ∩ B C ) : ι ( y ) = y } − { y ∈ Y ∩ S ′ ∩ V ( A C ′ ∩ B C ′ ) : ι ′ ( y ) = y } ⊆ ( V ( A C ′ ) − V ( B C ′ )) − ( V ( A C ) − V ( B C )) , ( C ′ , S ′ , T ′ ) satisfies (U3) and is useful.It is easy to see that V ( A C ′ ∩ B C ′ ) − S ′ ⊆ V ( A C ∩ B C ) − S . Note that each vertex v ∈ V ( A C ′ ∩ B C ′ ) − S ′ belongs to V ( A C ∩ B C ) − V ( A x ) , so N G ( v ) ∩ V ( A C ′ ) − V ( B C ′ ) = N G ( v ) ∩ V ( A C ) − V ( B C ) . Furthermore, S ′ − V ( A C ′ ∩ B C ′ ) ⊆ S − V ( A C ∩ B C ) . Hence every ( C ′ , S ′ , T ′ ) -heavy vertex is ( C , S, T ) -heavy. Since x is ( C , S, T ) -heavy but not ( C ′ , S ′ , T ′ ) -heavy,the set of ( C ′ , S ′ , T ′ ) -heavy vertices is strictly contained in the set of ( C , S, T ) -heavy vertices.Let v be a ( C ′ , S ′ , T ′ ) -dangerous vertex and let v ′ be a vertex in N G ( v ) ∩ V ( A C ′ ) − ( V ( B C ′ ) ∩ S ′ ) witnessing the definition of being dangerous. Since v S ′ , v ∈ V ( A C ∩ B C ) − V ( A x ) , so133 ′ ∈ N G [ V ( A C ∩ B C )] ∩ V ( A C ) ∩ V ( B x ) . So v ′ ∈ V ( B x ) − ( V ( B C ) ∪ S ′ ) . Since v ∈ V ( A C ′ ∩ B C ′ ) , v ′ ∈ N G ( v ) ∩ N G [ V ( A C ′ ∩ B C ′ )] . Since v ′ S ′ and v ′ V ( A C ∩ B C ) − V ( A C ′ ∩ B C ′ ) and v ′ ∈ V ( B x ) and N G ( v ′ ) ∩ V ( A C ∩ B C ) − V ( A x ) = ∅ , we know v ′ S by the procedure of modifying S . So v ′ ∈ ( N G ( v ) ∩ V ( A C ) − ( V ( B C ) ∪ S )) ∩ V ( B x ) . Note that T − T ′ ⊆ V ( A x ) − V ( B x ) . So if v ′ ∈ Y − T ′ , then v ′ ∈ Y − T and v is ( C , S, T ) -dangerous. Furthermore, Y ∩ V ( A C ) ∩ N G ( v ′ ) = Y ∩ V ( A C ′ ) ∩ N G ( v ′ ) and S ′ − V ( A C ′ ∩ B C ′ ) ⊆ S − V ( A C ∩ B C ) , so v is ( C , S, T ) -dangerous.Therefore, the set of ( C ′ , S ′ , T ′ ) -dangerous vertices is a subset of the set of ( C , S, T ) -dangerousvertices. This proves the claim. Claim 6.7.8. Let ( C , S, T ) be a useful triple. Then there exists a set S ′ with S ∪ ( Y ∩ V ( A C ∩ B C )) ⊆ S ′ ⊆ N G [ V ( A C ∩ B C )] ∩ V ( A C ) such that ( C , S ′ , T ) is a useful triple and: • If ι ′ is the bijection witnessing that ( C , S ′ , T ) satisfies (U3), then for every y ∈ Y ∩ V ( A C ∩ B C ) , the unique element of the domain of ι ′ mapped to y by ι ′ is y . • The set of ( C , S ′ , T ) -dangerous vertices is contained in the set of ( C , S, T ) -dangerousvertices. • The set of ( C , S ′ , T ) -heavy vertices is contained in the set of ( C , S, T ) -heavy vertices.Proof. Let ι be a function mentioned in (U3) witnessing that ( C , S, T ) is a useful triple. Wemay assume that Y ∩ V ( A C ∩ B C ) ⊆ S , since if some vertex y ∈ Y ∩ V ( A C ∩ B C ) does notbelong to S , then y is not in the domain of ι , and we can define ι ( y ) = y without violating(U3) and (U4) such that the set of dangerous vertices and the set of heavy vertices remainthe same.Since ι is a bijection, we write the element mapped to y by ι as ι ( − ( y ) . Modify ι and S by applying the following operations to some vertex y ∈ Y ∩ S ∩ V ( A C ∩ B C ) with ι ( − ( y ) = y ,and repeat until no such y exists: • remove ι ( − ( y ) from the domain of ι , • define ι ( y ) := y ,Then define S ′ and ι ′ to be the modified S and ι , respectively. Clearly, ( C , S ′ , T ) satisfies(U3), S ⊆ S ′ ⊆ N G [ V ( A C ∩ B C )] ∩ V ( A C ) , and ι ′ ( y ) = y for every y ∈ Y ∩ S ′ ∩ V ( A C ∩ B C ) .Since we assume that Y ∩ V ( A C ∩ B C ) ⊆ S , we have S ∪ ( Y ∩ V ( A C ∩ B C )) ⊆ S ′ ⊆ N G [ V ( A C ∩ B C )] ∩ V ( A C ) and ι ′ ( y ) = y for every y ∈ Y ∩ V ( A C ∩ B C ) . Since T ⊆ V ( A C ) − V ( B C ) , ( C , S ′ , T ) satisfies (U4). Since S ′ − S ⊆ V ( A C ∩ B C ) , the set of ( C , S ′ , T ) -dangerous verticesis contained in the set of ( C , S, T ) -dangerous vertices, and the set of ( C , S ′ , T ) -heavy verticesis contained in the set of ( C , S, T ) -heavy vertices. Claim 6.7.9. Let ( C , S, T ) be a useful triple, and let x be a ( C , S, T ) -dangerous vertex. Ifthere exists no ( C , S, T ) -heavy vertex, then there exists a useful triple ( C ′ , S ′ , T ′ ) with C ′ = C ∪ { ( A x , B x ) } such that the set of ( C ′ , S ′ , T ′ ) -dangerous vertices is strictly contained in theset of ( C , S, T ) -dangerous vertices.Proof. By Claim 6.7.8, we may assume that Y ∩ V ( A C ∩ B C ) ⊆ S and the function ι mentionedin (U3) witnessing that ( C , S, T ) is useful satisfies ι ( y ) = y for every y ∈ Y ∩ V ( A C ∩ B C ) .Let C ′ := C ∪ { ( A x , B x ) } .We first assume that | Y ∩ V ( A x ∩ B C ) | > d − . So | Y ∩ V ( A x ∩ B C ) | > | V ( A x ∩ B x ) ∩ V ( B C ) | by Claim 6.7.5. Hence there exists a function ι ′ whose domain is a subset of Y ∩ V ( A C ∪ A x ) such that: 134 ι ′ ( y ) = ι ( y ) for every y ∈ Y ∩ V ( A C ) − V ( A x ∩ B C ) belonging to the domain of ι with ι ( y ) ∈ V ( A C ) ∩ N G [ V ( A C ′ ∩ B C ′ )] − V ( A x ) , and • for each vertex v in V ( A x ∩ B x ) ∩ V ( B C ) , there exists exactly one element y ∈ Y ∩ V ( A x ∩ B C ) such that ι ′ ( y ) = v and if v ∈ Y , then y = v .Let S ′ := ( S ∩ N G [ V ( A C ′ ∩ B C ′ )] − V ( A x )) ∪ V ( A x ∩ B x ∩ B C ) . So ι ′ is a bijection from asubset of Y ∩ V ( A C ∪ A x ) to S ′ . Note that every vertex in S ′ − V ( A C ′ ∩ B C ′ ) is contained in S − ( V ( A x ) ∪ V ( A C ′ ∩ B C ′ )) , so it is adjacent to some vertex in V ( A C ′ ∩ B C ′ ) .Let T ′ := T . Since ι satisfies (U3) and ι ( y ) = y for every y ∈ Y ∩ S ∩ V ( A C ∩ B C ) , weknow that ι ′ satisfies (U3). Since T ′ = T ⊆ V ( A C ) − V ( B C ) , ( C ′ , S ′ , T ′ ) satisfies (U4). So ( C ′ , S ′ , T ′ ) is a useful triple.Let v be a ( C ′ , S ′ , T ′ ) -dangerous vertex. So v ∈ V ( A C ′ ∩ B C ′ ) − S ′ ⊆ V ( A C ∩ B C ) − V ( A x ) .Let v ′ be a vertex witnessing that v is ( C ′ , S ′ , T ′ ) -dangerous. So v ′ ∈ V ( B x ) − ( V ( B C ) ∪ S ′ ) and N G ( v ′ ) ⊆ V ( A C ) . Since v ∈ V ( A C ′ ∩ B C ′ ) − V ( A x ) , v ′ ∈ N G [ V ( A C ′ ∩ B C ′ )] , so v ′ S . Since S ′ − V ( A C ′ ∩ B C ′ ) ⊆ S − V ( A C ∩ B C ) , if v ′ Y and | (( Y ∩ V ( A C ′ )) ∪ ( S ′ − V ( A C ′ ∩ B C ′ ))) ∩ N G ( v ′ ) | > d − , then v ′ Y and | (( Y ∩ V ( A C )) ∪ ( S − V ( A C ∩ B C ))) ∩ N G ( v ′ ) | > d − ,so v is ( C , S, T ) -dangerous. Since T ′ = T , if v ′ ∈ Y − T ′ , then v ′ ∈ Y − T and v is ( C , S, T ) -dangerous. So every ( C ′ , S ′ , T ′ ) -dangerous vertex is ( C , S, T ) -dangerous. Since x is ( C , S, T ) -dangerous but not ( C ′ , S ′ , T ′ ) -dangerous, the set of ( C ′ , S ′ , T ′ ) -dangerous vertices isstrictly contained in the set of ( C , S, T ) -dangerous vertices. So the claim holds.Hence we may assume that | Y ∩ V ( A x ∩ B C ) | d − .Modify S and define ι ′ to be the function obtained from ι by applying the followingoperations to a vertex y in the domain of ι with ι ( y ) N G [ V ( A C ′ ∩ B C ′ )] ∩ V ( A C ′ ) , andrepeating until no such y exists: • if y ∈ V ( A C ∩ B C ) − V ( B x ) or V ( A x ∩ B x ) ∩ V ( B C ) − S = ∅ , then remove y from thedomain of ι and remove ι ( y ) from S , • if y V ( A C ∩ B C ) − V ( B x ) and N G ( y ) ∩ V ( A C ′ ∩ B C ′ ) − S = ∅ and V ( A x ∩ B x ) ∩ V ( B C ) − S = ∅ , then redefine ι ( y ) to be an element in V ( A x ∩ B x ) ∩ V ( B C ) − S and add this elementinto S , • otherwise remove ι ( y ) from S , redefine ι ( y ) to be an element in N G ( y ) ∩ V ( A C ′ ∩ B C ′ ) − S and add this element into S .Let S ′ be the modified S , and let T ′ := T ∪ ( Y ∩ V ( B C ) − V ( A C ∪ B x )) ∪ ( Y ∩ V ( A C ∩ B C ) − ( V ( B x ) ∪ N G [ V ( A C ′ ∩ B C ′ )])) . Clearly, T ′ is disjoint from the domain of ι ′ . By (U2), Z ′ ∩ V ( B C ) = ∅ . So T ′ is disjointfrom Z ′ as T is disjoint from Z ′ . So ( C ′ , S ′ , T ′ ) satisfies (U4).Since Y ∩ V ( A C ∩ B C ) ⊆ S , we know Y ∩ V ( A C ∩ B C ) − ( V ( B x ) ∪ N G [ V ( A C ′ ∩ B C ′ )]) ⊆ S .For every y ∈ Y ∩ V ( A C ∩ B C ) − ( V ( B x ) ∪ N G [ V ( A C ′ ∩ B C ′ )]) , since ι ( y ) = y N G [ V ( A C ′ ∩ B C ′ )] ∩ V ( A C ′ ) and y ∈ V ( A C ∩ B C ) − V ( B x ) , y ∈ S − S ′ .So | S | > | S ′ | + | Y ∩ V ( A C ∩ B C ) − ( V ( B x ) ∪ N G [ V ( A C ′ ∩ B C ′ )]) | . Hence | S ′ | + | T ′ | | S | + | T | + | Y ∩ V ( B C ) − V ( A C ∪ B x ) | . Since Y ∩ V ( B C ) − V ( A C ∪ B x ) ⊆ Y ∩ ( V ( A C ′ ) − V ( B C ′ )) − V ( A C ) and ( C , S, T ) satisfies (U3), we know | S ′ | + | T ′ | + | Z | + 1 | S | + | T | + | Y ∩ V ( B C ) − V ( A C ∪ B x ) | + | Z | + 1 | Y ∩ V ( A C ) − V ( B C ) | + |{ y ∈ Y ∩ S ∩ V ( A C ∩ B C ) : ι ( y ) = y }| + | Y ∩ ( V ( A C ′ ) − V ( B C ′ )) − V ( A C ) | | Y ∩ V ( A C ′ ) − V ( B C ′ ) | − | Y ∩ V ( A C ∩ B C ) − V ( B x ) | + |{ y ∈ Y ∩ S ∩ V ( A C ∩ B C ) : ι ( y ) = y }| | Y ∩ V ( A C ′ ) − V ( B C ′ ) | + |{ y ∈ Y ∩ S ∩ V ( A C ∩ B C ) ∩ V ( B x ) : ι ( y ) = y }| | Y ∩ V ( A C ′ ) − V ( B C ′ ) | + |{ y ∈ Y ∩ S ′ ∩ V ( A C ′ ∩ B C ′ ) : ι ′ ( y ) = y }| . Hence ( C ′ , S ′ , T ′ ) satisfies (U3). Therefore ( C ′ , S ′ , T ′ ) is useful.Suppose that the set of ( C ′ , S ′ , T ′ ) -dangerous vertices is not strictly contained in the setof ( C , S, T ) -dangerous vertices. Since x is ( C , S, T ) -dangerous but not ( C ′ , S ′ , T ′ ) -dangerous,there exists a vertex v that is ( C ′ , S ′ , T ′ ) -dangerous but not ( C , S, T ) -dangerous. So there existsa vertex v ′ ∈ N G ( v ) ∩ V ( A C ′ ) − ( V ( B C ′ ) ∪ S ′ ) such that either v ′ ∈ Y − T ′ , or v ′ Y and | (( Y ∩ V ( A C ′ )) ∪ ( S ′ − V ( A C ′ ∩ B C ′ )) ∩ N G ( v ′ ) | > d − . Since v ′ ∈ N G [ V ( A C ′ ∩ B C ′ )] ∩ V ( A C ′ ) ,if v ′ belongs to S at beginning, then v ′ is not removed from S during the process of modifying S , so v ′ ∈ S ′ , a contradiction. So v ′ S .Suppose that v ′ ∈ V ( A C ) − V ( B C ) . So v ∈ V ( A C ∩ B C ) ∩ V ( A C ′ ∩ B C ′ ) ⊆ V ( A C ∩ B C ) ∩ V ( B x ) .Hence if v belongs to S at beginning, then v is not removed from S during the process ofmodifying S , so v ∈ S ′ . Since v is ( C ′ , S ′ , T ′ ) -dangerous, v S ′ , so v S . Since v is not ( C , S, T ) -dangerous, v ′ Y − T , and either v ′ ∈ Y or | (( Y ∩ V ( A C )) ∪ ( S − V ( A C ∩ B C ))) ∩ N G ( v ′ ) | < d − . Since T ′ ∩ V ( A C ) − V ( B C ) = T ∩ V ( A C ) − V ( B C ) , v ′ Y − T ′ . Since v is ( C ′ , S ′ , T ′ ) -dangerous, v ′ Y and | (( Y ∩ V ( A C ′ )) ∪ ( S ′ − V ( A C ′ ∩ B C ′ ))) ∩ N G ( v ′ ) | > d − .Since N G ( v ′ ) ⊆ V ( A C ) and S ′ − V ( A C ′ ∩ B C ′ ) ⊆ S − V ( A C ∩ B C ) , d − | (( Y ∩ V ( A C ′ )) ∪ ( S ′ − V ( A C ′ ∩ B C ′ ))) ∩ N G ( v ′ ) | = | (( Y ∩ V ( A C )) ∪ ( S ′ − V ( A C ′ ∩ B C ′ ))) ∩ N G ( v ′ ) | | (( Y ∩ V ( A C )) ∪ ( S − V ( A C ∩ B C ))) ∩ N G ( v ′ ) | < d − , a contradiction.Therefore, v ′ ∈ V ( B C ) . So v ′ ∈ ( V ( A x ) − V ( B x )) ∩ V ( B C ) and hence N G ( v ′ ) ⊆ V ( A x ) .Since Y ∩ V ( A C ∩ B C ) ⊆ S , if v ′ ∈ Y ∩ V ( A C ∩ B C ) , then v ′ ∈ S , a contradiction. So v ′ Y ∩ V ( A C ∩ B C ) . Since Y ∩ V ( B C ) − V ( A C ∪ B x ) ⊆ T ′ , if v ′ ∈ Y − T ′ , then v ′ ∈ Y ∩ V ( A C ∩ B C ) , a contradiction. So v ′ Y − T ′ . Since v is ( C ′ , S ′ , T ′ ) -dangerous, | (( Y ∩ V ( A C ′ )) ∪ ( S ′ − V ( A C ′ ∩ B C ′ ))) ∩ N G ( v ′ ) | > d − . Since | Y ∩ V ( A x ∩ B C ) | d − and ( S ′ − V ( A C ′ ∩ B C ′ )) − ( V ( A C ) ∪ V ( B x )) = ∅ and N G ( v ′ ) ⊆ V ( A x ) , | N G ( v ′ ) ∩ (( Y ∩ V ( A C ) − V ( B C )) ∪ ( S − V ( A C ∩ B C )) | > | N G ( v ′ ) ∩ (( Y ∩ V ( A x ) − V ( B C )) ∪ ( S ′ − V ( A C ′ ∩ B C ′ )) | > | N G ( v ′ ) ∩ (( Y ∩ V ( A x )) ∪ ( S ′ − ( V ( A C ′ ∩ B C ′ )))) | − ( d − | N G ( v ′ ) ∩ ( Y ∪ ( S ′ − ( V ( A C ′ ∩ B C ′ )))) | − ( d − > (2 d − − ( d − d − . v ′ ∈ V ( A C ∩ B C ) − V ( B x ) . Since there exists no ( C , S, T ) -heavy vertex, v ′ is nota ( C , S, T ) -heavy vertex. So v ′ ∈ S , a contradiction. This proves the claim. Claim 6.7.10. If ( C , S, T ) is a useful triple such that there exists a ( C , S, T ) -dangerous vertex,then there exists a useful triple ( C ′ , S ′ , T ′ ) with C ⊆ C ′ and |C ′ | |C| + | V ( A C ∩ B C ) | +1 such thatthe set of ( C ′ , S ′ , T ′ ) -dangerous vertices is strictly contained in the set of ( C , S, T ) -dangerousvertices.Proof. Note that there are at most | V ( A C ∩ B C ) | ( C , S, T ) -heavy vertices. By repeatedlyapplying Claim 6.7.7 at most | V ( A C ∩ B C ) | times, there exists a useful triple ( C , S , T ) with C ⊆ C and |C | |C| + | V ( A C ∩ B C ) | such that there exists no ( C , S , T ) -heavy vertices,and the set of ( C , S , T ) -dangerous vertices is contained in the set of ( C , S, T ) -dangerousvertices. By Claim 6.7.9 applied to C , there exists a useful triple ( C ′ , S ′ , T ′ ) with C ⊆ C ′ and |C ′ | = |C | + 1 |C| + | V ( A C ∩ B C ) | + 1 such that the set of ( C ′ , S ′ , T ′ ) -dangerous vertices isstrictly contained in the set of ( C , S , T ) -dangerous vertices and hence is strictly containedin the set of ( C , S, T ) -dangerous vertices. This proves the claim. Claim 6.7.11. There exists a useful triple ( C ∗ , S ∗ , T ∗ ) with |C ∗ | a ( d − a such that thereexists no ( C ∗ , S ∗ , T ∗ ) -dangerous vertex.Proof. By Claim 6.7.6, there exists a useful triple ( C , ∅ , ∅ ) with |C | | Z ′ | d + | Z | + 1 .Let S = ∅ and T = ∅ . So ( C , S , T ) is a useful triple with |C | f ( ξ ) d + ξ + 1 = a .For i > , if there exists a ( C i − , S i − , T i − ) -dangerous vertex, then by Claim 6.7.10, thereexists a useful triple ( C i , S i , T i ) such that |C i | |C i − | + | V ( A C i − ∩ B C i − ) | + 1 and the setof ( C i , S i , T i ) -dangerous vertices is strictly contained in the set of ( C i − , S i − , T i − ) -dangerousvertices. So |C i | a i − + ( d − a i − + 1 a i for each i > by induction on i . Sincethere are at most | V ( A C ∩ B C ) | |C | ( d − ( d − a ( C , S , T ) -dangerous vertices.Hence there exists i ∗ with i ∗ ( d − a such that ( C i ∗ , S i ∗ , T i ∗ ) is a useful triple with no ( C i ∗ , S i ∗ , T i ∗ ) -dangerous vertex. Note that |C i ∗ | a i ∗ a ( d − a .Let ι ∗ be the function mentioned in (U3) witnessing that ( C ∗ , S ∗ , T ∗ ) is useful. ByClaim 6.7.8, we may assume that Y ∩ V ( A C ∗ ∩ B C ∗ ) ⊆ S ∗ such that ι ∗ ( y ) = y for every y ∈ Y ∩ V ( A C ∗ ∩ B C∗ ) .Define the following: G B := G [( \ ( A,B ) ∈C ∗ V ( B )) ∪ ( Z ∪ S ∗ ∪ T ∗ )] Y B := ( Y ∩ V ( B C ∗ )) ∪ ( Z ∪ S ∗ ∪ T ∗ ) . Claim 6.7.12. For every vertex v ∈ V ( G B ) − Y B , N G ( v ) ∩ Y ⊆ N G B ( v ) ∩ Y B .Proof. Suppose to the contrary that there exist v ∈ V ( G B ) − Y B and y ∈ N G ( v ) ∩ Y − ( N G B ( v ) ∩ Y B ) . Since y ∈ Y − Y B , y ∈ V ( A C ∗ ) − V ( B C ∗ ) . So v ∈ V ( A C ∗ ∩ B C ∗ ) − S ∗ . Sincethere exists no ( C ∗ , S ∗ , T ∗ ) -dangerous vertex, v is not a ( C ∗ , S ∗ , T ∗ ) -dangerous vertex. Since y ∈ N G ( v ) ∩ V ( A C ∗ ) − ( V ( B C ∗ ) ∪ S ∗ ) , y Y − T ∗ . Since y ∈ Y , y ∈ T ∗ . So y ∈ Y B , acontradiction.Define the following: 137 For every y ∈ Y B , let L B ( y ) be a 1-element subset of L ( y ) . • For every v ∈ V ( G B ) − Y B with | N G B ( v ) ∩ Y B | ∈ [ s ′ − , let L B ( v ) be a subset of L ( v ) with size s ′ + 2 − | N G B ( v ) ∩ Y B | such that L B ( v ) ∩ L B ( u ) = ∅ for every u ∈ N G ( v ) ∩ Y B .(Note that such a subset of L ( v ) exists by Claim 6.7.12.) • For every other vertex v of G B , let L B ( v ) := L ( v ) .Hence L B is a restricted ( s ′ , , Y B ) -list-assignment by Claim 6.7.12. Since ( C ∗ , S ∗ , T ∗ ) is usefuland ι ∗ ( y ) = y for every y ∈ Y ∩ V ( A C ∗ ∩ B C∗ ) , | Y B | | Y ∩ V ( B C ∗ ) | + | S ∗ | − | Y ∩ V ( A C ∗ ∩ B C ∗ ) | + | Z | + | T ∗ | ( | Y | − | Y ∩ V ( A C ∗ ) − V ( B C ∗ ) | ) + | S ∗ | − | Y ∩ V ( A C ∗ ∩ B C ∗ ) | + | Z | + | T ∗ | | Y | − by (U3).Since | Y B | < | Y | , we know | V ( G B ) | < | V ( G ) | . By the minimality of | V ( G ) | , there existsan ( η, g ) -bounded L B -coloring c B of G B . Define the following: • Let G A := G [ V ( A C ∗ ) ∪ Z ] . • Let Y A := ( Y ∩ V ( A C ∗ )) ∪ Z ∪ S ∗ ∪ T ∗ ∪ V ( A C ∗ ∩ B C ∗ ) . • For every y ∈ Y A , let L A ( y ) be a 1-element subset of L ( y ) such that if y ∈ V ( G B ) , then L A ( y ) = { c B ( y ) } . • For every v ∈ V ( G A ) − Y A with | N G A ( v ) ∩ Y A | s ′ − , let L A ( v ) be a subset of L ( v ) with size s ′ + 2 − | N G A ( v ) ∩ Y A | such that L A ( v ) ∩ L A ( u ) = ∅ for every u ∈ Y A ∩ N G A ( v ) . • For every other vertex v of G A , let L A ( v ) := L ( v ) .Then L A is a restricted ( s ′ , , Y A ) -list-assignment of G A . Since |C ∗ | a ( d − a , | V ( A C ∗ ∩ B C ∗ ) | ( d − a ( d − a < θ − ξ . So ( A C ∗ , B C ∗ ) ∈ T − Z and hence | Y ∩ V ( A C ∗ ) | θ . By (U3), | S ∗ | + | T ∗ | + | Z | | Y ∩ V ( A C ∗ ) − V ( B C ∗ ) | + | Y ∩ V ( A C ∗ ∩ B C ∗ ) | | Y ∩ V ( A C ∗ ) | θ . Hence | Y A | | Y ∩ V ( A C ∗ ) | + | S ∗ | + | T ∗ | + | Z | + | V ( A C ∗ ∩ B C ∗ ) | θ + 3 θ + θ θ .In particular, | V ( G A ) | < | V ( G ) | . By minimality, there exists an ( η, g ) -bounded L A -coloring c A of G A . Claim 6.7.13. For every v ∈ V ( G A ) − ( V ( G B ) ∪ Y ) with N G ( v ) ∩ V ( A C ∗ ∩ B C ∗ ) − S ∗ = ∅ , c B ( u ) L A ( v ) for every u ∈ N G ( v ) ∩ V ( G B ) .Proof. Since v ∈ V ( G A ) − ( V ( G B ) ∪ Y ) , v ∈ V ( G ) − ( Z ∪ V ( B C ∗ )) . So there exists ( A, B ) ∈ C ∗ such that v ∈ V ( A ) − V ( B ) . Hence N G ( v ) ⊆ V ( A ) and | N G ( v ) ∩ V ( A C ∗ ∩ B C ∗ ) | | N G ( v ) ∩ V ( A ∩ B ) | d − .Since v ∈ N G ( A C ∗ ∩ B C ∗ ) and C ∗ satisfies (U2), v Z ′ . So | N G ( v ) ∩ Z | s − .Since N G ( v ) ∩ V ( A C ∗ ∩ B C ∗ ) − S ∗ = ∅ , there exists w ∈ N G ( v ) ∩ V ( A C ∗ ∩ B C ∗ ) − S ∗ . Sincethere exists no ( C ∗ , S ∗ , T ∗ ) -dangerous vertex, w is not a ( C ∗ , S ∗ , T ∗ ) -dangerous vertex. Since S ∗ ⊆ V ( G B ) , v S ∗ . Since v Y , v Y − T ∗ . So | N G ( v ) ∩ (( Y ∩ V ( A C ∗ )) ∪ ( S ∗ − V ( A C ∗ ∩ B C ∗ ))) | d − .Since T ∗ ⊆ Y ∩ V ( A C ∗ ) , | N G ( v ) ∩ Y A | | N G ( v ) ∩ V ( A C ∗ ∩ B C ∗ ) | + | N G ( v ) ∩ Z | + | N G ( v ) ∩ (( Y ∩ V ( A C ∗ )) ∪ ( S ∗ − V ( A C ∗ ∩ B C ∗ ))) | ( d − 1) + ( s − 1) + (2 d − d + s − s ′ − . So by the definition of L A , L A ( v ) ∩ { c B ( u ) } = L A ( v ) ∩ L A ( u ) = ∅ for every u ∈ Y A ∩ V ( G B ) ∩ N G ( v ) . Since N G ( v ) ∩ V ( G B ) ⊆ Y A , c B ( u ) L A ( v ) for every u ∈ N G ( v ) ∩ V ( G B ) .Let c be the L -coloring of G defined by c ( v ) := c A ( v ) if v ∈ V ( G A ) , and c ( v ) := c B ( v ) if v ∈ V ( G ) − V ( G A ) . Claim 6.7.14. Let M be a c -monochromatic component intersecting both V ( G A ) − V ( G B ) and V ( G B ) − V ( G A ) . Then every component of M ∩ G A intersects Y A , and every componentof M ∩ G B intersects Y B .Proof. Since M intersects both V ( G A ) − V ( G B ) and V ( G B ) − V ( G A ) , every component of M ∩ G A intersects V ( A C ∗ ∩ B C ∗ ) ∪ Z ∪ S ∗ ∪ T ∗ ⊆ Y A .Let M B be a component of M ∩ G B . Suppose that M B is disjoint from Y B = ( Y ∩ V ( B C ∗ )) ∪ Z ∪ S ∗ ∪ T ∗ . Since M intersects both V ( G A ) − V ( G B ) and V ( G B ) − V ( G A ) , thereexist u ∈ V ( M B ) ∩ V ( A C ∗ ∩ B C ∗ ) − S ∗ and v ∈ N M B ( v ) ∩ V ( A C ∗ ) − ( V ( B C ∗ ) ∪ S ∗ ∪ T ∗ ) .Since u is not a ( C ∗ , S ∗ , T ∗ ) -dangerous vertex, v Y − T ∗ . Since v T ∗ , v Y . So v ∈ V ( G A ) − ( V ( G B ) ∪ Y ) and N G ( v ) ∩ V ( A C ∗ ∩ B C ∗ ) − S ∗ ⊇ { u } 6 = ∅ . By Claim 6.7.13, c ( v ) = c A ( v ) = c B ( u ) = c ( u ) . But M is a c -monochromatic component, a contradiction.Hence every component of M ∩ G B intersects Y B .Let U A be the union of the c A -monochromatic components of G A intersecting Y A . Let U B be the union of the c B -monochromatic components of G B intersecting Y B . Since c A and c B are ( η, g ) -bounded, | V ( U A ) ∪ V ( U B ) | | Y A | g ( | Y A | ) + | Y B | g ( | Y B | ) (7 θ ) g (7 θ ) + ( | Y | − g ( | Y | − g ( | Y | ) .Since V ( G ) ⊆ V ( G A ) ∪ V ( G B ) , by Claim 6.7.14, every c -monochromatic component in-tersecting both V ( G A ) − V ( G B ) and V ( G B ) − V ( G A ) is contained in U A ∪ U B and hencecontains at most g ( | Y | ) η g ( η ) vertices. Let M be a c -monochromatic component. If V ( M ) ⊆ V ( G A ) , then M is a c A -monochromatic component with at most η g ( η ) verticessince c A is ( η, g ) -bounded. If V ( M ) ⊆ V ( G B ) , then M is a c B -monochromatic componentwith at most η g ( η ) vertices since c B is ( η, g ) -bounded. Hence every c -monochromatic com-ponent contains at most η g ( η ) vertices.Since Y ⊆ Y A ∪ Y B , by Claim 6.7.14, the union of the c -monochromatic componentsintersecting Y is contained in U A ∪ U B , so it contains at most g ( | Y | ) | Y | g ( | Y | ) vertices.Therefore, c is an ( η, g ) -bounded L -coloring of G , a contradiction. This proves the theorem. ( -Subdivisions Recall that an almost ( -subdivision of a graph H is a graph obtained from H bysubdividing edges such that at most one edge is subdivided more than once. The followingsimple observation is useful. Lemma 6.8. For s ∈ N , let G be a graph and let H be a subgraph of G isomorphic to K s − ,t for some t > (cid:0) s − (cid:1) + 2 . Let ( X, Z ) be the bipartition of H with | X | = s − . If G does notcontain an almost ( -subdivision of K s +1 , then each component of G − X contains at mostone vertex in Z , and G − X has at least two components. roof. Let C , C , . . . , C k be the components of G − X . For each i ∈ [ k ] , | V ( C i ) ∩ Z | ,as otherwise G [ X ∪ Z ] together with a path in C i connecting two vertices in V ( C i ) ∩ Z is analmost ( -subdivision of K s +1 , a contradiction. Hence k > | Z | > t > .The following lemma shows that a result for graphs excluding a K s,t -subgraph can beextended for graphs excluding an almost ( -subdivision of K s +1 . Let s, r ∈ N . Let G bea graph and Y ⊆ V ( G ) . An ( s, r, Y ) -list-assignment of G is said to be faithful if for every v ∈ V ( G ) − Y with | N G ( v ) ∩ Y | = s , we have L ( v ) − S y ∈ Y ∩ N G ( v ) L ( y ) = ∅ . Lemma 6.9. Let G be a subgraph-closed family of graphs. Let β, r be functions with domain N such that β ( x ) > x and r ( x ) ∈ N for every x ∈ N .Assume that for every s ∈ N , there exists η ∈ N and a nondecreasing function g suchthat for every G ∈ G with no K s,t s -subgraph, where t s := max { (cid:0) s (cid:1) + 2 , s + 2 } , for every Y ⊆ V ( G ) with | Y | η , and for every ( β ( s ) , r ( s ) , Y ) -list-assignment L of G , there existsan ( η, g ) -bounded L -coloring of G .Then for every s ∈ N with s > , there exists η ∗ ∈ N and a nondecreasing function g ∗ such that for every graph G ∈ G containing no almost ( -subdivision of K s +1 , for every Y ⊆ V ( G ) with | Y | η ∗ , and for every faithful ( β ( s − , r ( s − , Y ) -list-assignment L of G , there exists an ( η ∗ , g ∗ ) -bounded L -coloring of G .Proof. For every s ∈ N , let η s be the number and g s be the function such that for every G ∈ G with no K s,t s -subgraph, every Y ⊆ V ( G ) with | Y | η s and every ( β ( s ) , r ( s ) , Y ) -list-assignment of G , there exists an ( η s , g s ) -bounded L -coloring of G . For every s ∈ N with s > , let η ∗ s := η s − and let g ∗ s be the function defined by g ∗ s (0) := g s − (0) and g ∗ s ( x ) := g s − ( x ) + η ∗ s · g ∗ s ( x − for every x ∈ N .Fix s ∈ N −{ } . Let β ′ := β ( s − and r ′ := r ( s − . We shall prove that for every graph G in G containing no almost ( -subdivision of K s +1 , for every Y ⊆ V ( G ) with | Y | η ∗ s , andfor every faithful ( β ′ , r ′ , Y ) -list-assignment L of G , there exists an ( η ∗ s , g ∗ s ) -bounded L -coloringof G .Suppose to the contrary that G is a graph in G containing no almost ( -subdivision of K s +1 , Y is a subset of V ( G ) with | Y | η ∗ s , and L is a faithful ( β ′ , r ′ , Y ) -list-assignment of G such that there exists no ( η ∗ s , g ∗ s ) -bounded L -coloring of G . We further assume that | V ( G ) | is as small as possible.Since η ∗ s = η s − and g ∗ s > g s − , there exists no ( η s − , g s − ) -bounded L -coloring of G . Since η ∗ s = η s − , by the definition of η s − and g s − , G contains a K s − ,t s − -subgraph. Let t ′ be themaximum integer such that G contains a K s − ,t ′ -subgraph. So t ′ > t s − . Let H be a subgraphof G isomorphic to K s − ,t ′ . Let { P, Q } be the bipartition of H such that | P | = s − and | Q | = t ′ . By the maximality of t ′ , Q is the set of all vertices of V ( G ) − P adjacent in G to allvertices in P . Claim 6.9.1. Every component of G − P contains some vertex in Y .Proof. Suppose to the contrary that there exists a component C of G − P disjoint from Y .By Lemma 6.8, G − P contains at least two components and there exists at most one vertex in C adjacent in G to all vertices in P . By the minimality of G , there exists an ( η ∗ s , g ∗ s ) -bounded L | V ( G ) − V ( C ) -coloring c of G − V ( C ) . 140ince β ′ > s − and L is an ( β ′ , r ′ , Y ) -list-assignment and V ( C ) ∩ Y = ∅ , for every v ∈ V ( C ) with | N G ( v ) ∩ Y | β ′ − , we have L ( v ) ∩ S y ∈ N G ( v ) ∩ Y L ( y ) = ∅ and | L ( v ) − { c ( y ) : y ∈ N G ( v ) ∩ P }| > | L ( v ) | − |{ c ( y ) : y ∈ N G ( v ) ∩ P − Y }| = β ′ + r ′ − | N G ( v ) ∩ Y | − |{ c ( y ) : y ∈ N G ( v ) ∩ P − Y }| = β ′ + r ′ − | N G ( v ) ∩ Y ∩ P | − |{ c ( y ) : y ∈ N G ( v ) ∩ P − Y }| > β ′ + r ′ − | N G ( v ) ∩ P | > β ′ + r ′ − ( s − > . For every v ∈ V ( C ) with | N G ( v ) ∩ Y | > β ′ , | N G ( v ) ∩ P | > | N G ( v ) ∩ Y | > β ′ = β ( s − > s − , implying P ⊆ Y and | N G ( v ) ∩ Y | = β ′ , so L ( v ) − { c ( y ) : y ∈ N G ( v ) ∩ P } = L ( v ) − S y ∈ N G ( v ) ∩ Y L ( y ) = ∅ since L is faithful. So for every v ∈ V ( C ) , L ( v ) − { c ( y ) : y ∈ N G ( v ) ∩ P } 6 = ∅ .Let L ′ be the following list-assignment of G [ V ( C ) ∪ P ] : • For every v ∈ P , let L ′ ( v ) := { c ( v ) } . • For every v ∈ V ( C ) with | N G ( v ) ∩ P | > β ′ , let L ′ ( v ) be a 1-element subset of L ( v ) − S u ∈ P L ′ ( u ) . • Let Y ′ := P ∪ { v ∈ V ( C ) : | N G ( v ) ∩ P | > β ′ } . • For every v ∈ V ( C ) ∩ N <β ′ ( Y ′ ) , let L ′ ( v ) be a subset of L ( v ) − S y ∈ N G ( v ) ∩ Y ′ L ′ ( y ) of size | L ( v ) | − | N G ( v ) ∩ Y ′ − Y | = β ′ + r ′ − | N G ( v ) ∩ Y ′ | > r ′ + 1 . • For every v ∈ V ( C ) ∩ N <β ′ ( P ) − N <β ′ ( Y ′ ) , let L ′ ( v ) be a subset of L ( v ) − S y ∈ N G ( v ) ∩ P L ′ ( y ) of size | L ( v ) | − | N G ( v ) ∩ P − Y | = β ′ + r ′ − | N G ( v ) ∩ P | > r ′ + 1 . • For every v ∈ V ( C ) − ( Y ′ ∪ N <β ′ ( Y ′ ) ∪ N <β ′ ( P )) , let L ′ ( v ) := L ( v ) .Note that Y ′ − P consists of the vertex in V ( C ) adjacent in G to all vertices in P . Hencefor every v ∈ V ( C ) ∩ N G ( P ) − Y ′ , | N G ( v ) ∩ P | ∈ [ β ′ − . That is, V ( C ) ∩ N G ( P ) − Y ′ = V ( C ) ∩ N <β ′ ( P ) . So for every v ∈ V ( C ) ∩ N G ( P ) − Y ′ , L ′ ( v ) ∩ L ′ ( u ) = ∅ for every u ∈ P ∩ N G ( v ) .Clearly, L ′ is an ( β ′ , r ′ , Y ′ ) -list-assignment of G . If v ∈ V ( C ) − Y ′ with | N G ( v ) ∩ Y ′ | = β ′ ,then since | Y ′ − P | = 1 , we know v ∈ N <β ′ ( P ) − N <β ′ ( Y ′ ) , so L ′ ( v ) is a set of size at least r ′ +1 > disjoint from S y ∈ N G ( v ) ∩ P L ′ ( y ) . Hence if v ∈ ( V ( C ) ∪ P ) − Y ′ with | N G ( v ) ∩ Y ′ | = β ′ ,then L ′ ( v ) − S y ∈ N G ( v ) ∩ Y ′ L ′ ( y ) = L ′ ( v ) − S y ∈ N G ( v ) ∩ Y ′ − P L ′ ( y ) has size | L ′ ( y ) |− > . Therefore, L ′ is a faithful ( β ′ , r ′ , Y ′ ) -list-assignment of G [ V ( C ) ∪ P ] .Since G − P contains at least two components, | V ( C ) ∪ P | < | V ( G ) | . By the minimalityof G , there exists an ( η ∗ s , g ∗ s ) -bounded L ′ -coloring c ′ of G [ V ( C ) ∪ P ] .For every v ∈ V ( C ) ∩ N G ( P ) , if v ∈ N <β ′ ( P ) , then L ′ ( v ) is disjoint from S y ∈ N G ( v ) ∩ P L ′ ( y ) ;if v ∈ V ( C ) with | N G ( v ) ∩ P | > β ′ , then v ∈ Y ′ − P and L ′ ( v ) is disjoint from S y ∈ P L ′ ( y ) .Hence every c ′ -monochromatic component intersecting P is contained in G [ P ] .Let c ∗ be the L -coloring defined by c ∗ ( v ) := c ( v ) if v ∈ V ( G ) − V ( C ) , and c ∗ ( v ) := c ′ ( v ) if v ∈ V ( C ) . Hence every c ∗ -monochromatic component is either contained in G − V ( C ) or141ontained in C , so it contains at most η ∗ s g ( η ∗ s ) vertices. Since V ( C ) ∩ Y = ∅ , the union ofthe c ∗ -monochromatic components intersecting Y equals the union of the c -monochromaticcomponents intersecting Y , and contains at most | Y | g ( | Y | ) vertices. Therefore, c ∗ is an ( η ∗ s , g ∗ s ) -bounded L -coloring of G , a contradiction.Let C , C , . . . , C k be the components of G − P . For i ∈ [ k ] , let G i := G [ V ( C i ) ∪ P ] .By Lemma 6.8, k > t s − > s + 1 . By Claim 6.9.1, Y ∩ V ( C i ) = ∅ for each i ∈ [ k ] ,so | V ( G i ) ∩ Y | | Y | − ( k − | Y | − s for each i ∈ [ k ] , and k | Y | η ∗ s . So | ( V ( G i ) ∩ Y ) ∪ P | | Y | − s + | P | < | Y | for each i ∈ [ k ] .Let L ∗ be the following list-assignment of G : • Let Y ∗ := Y ∪ P . • For each v ∈ Y ∗ , let L ∗ ( v ) be a 1-element subset of L ( v ) . • For each v ∈ N <β ′ ( Y ∗ ) , let L ∗ ( v ) be a subset of L ( v ) − S y ∈ N G ( v ) ∩ Y ∗ L ∗ ( y ) with size | L ( v ) | − | N G ( v ) ∩ ( Y ∗ − Y ) | = β ′ + r ′ − | N G ( v ) ∩ Y ∗ | . • For each v ∈ V ( G ) − ( Y ∗ ∪ N <β ′ ( Y ∗ )) , let L ∗ ( v ) := L ( v ) .Clearly, L ∗ is an ( β ′ , r ′ , Y ∗ ) -list-assignment of G . Let v ∈ V ( G ) − Y ∗ with | N G ( v ) ∩ Y ∗ | = β ′ . So L ∗ ( v ) = L ( v ) . If | N G ( v ) ∩ Y | = β ′ , then N G ( v ) ∩ Y ∗ = N G ( v ) ∩ Y , so L ∗ ( v ) − S y ∈ N G ( v ) ∩ Y ∗ L ∗ ( y ) = L ( v ) − S y ∈ N G ( v ) ∩ Y L ( y ) = ∅ since L is a faithful ( β ′ , r ′ , Y ) -list-assignment of G . If | N G ( v ) ∩ Y | < β ′ , then | L ∗ ( v ) | = | L ( v ) | = β ′ + r ′ − | N G ( v ) ∩ Y | and L ( v ) is disjoint from S y ∈ N G ( v ) ∩ Y L ( y ) , so | L ∗ ( v ) − [ y ∈ N G ( v ) ∩ Y ∗ L ∗ ( y ) | = | L ∗ ( v ) − [ y ∈ N G ( v ) ∩ Y ∗ − Y L ∗ ( y ) | > | L ∗ ( v ) | − | [ y ∈ N G ( v ) ∩ Y ∗ − Y L ∗ ( y ) | > β ′ + r ′ − | N G ( v ) ∩ Y | − | N G ( v ) ∩ Y ∗ − Y | = β ′ + r ′ − | N G ( v ) ∩ Y ∗ | = r > . Therefore, L ∗ is a faithful ( β ′ , r ′ , Y ∗ ) -list-assignment of G .Since P ⊆ Y ∗ , L ∗ | V ( G i ) is a faithful ( β ′ , r ′ , Y ∗ ∩ V ( G i )) -list-assignment of G i . Recall thatfor each i ∈ [ k ] , | Y ∗ ∩ V ( G i ) | | Y | − η . By the minimality of G , for each i ∈ [ k ] , thereexists an ( η ∗ s , g ∗ s ) -bounded L ∗ | V ( G i ) -coloring c i of G i . Since P ⊆ Y ∗ ∩ V ( G i ) for every i ∈ [ k ] ,we know for every v ∈ P , c i ( v ) = c j ( v ) for any i, j ∈ [ k ] . Let c ∗ be the L ∗ -coloring of G defined by c ( v ) := c ( v ) if v ∈ P , and c ( v ) := c i ( v ) if v ∈ V ( C i ) for some i ∈ [ k ] .Since P ⊆ Y ∗ ∩ V ( G i ) for all i ∈ [ k ] , the number of vertices in the union of the c ∗ -monochromatic components intersecting Y ∪ P is at most k X i =1 | Y ∗ ∩ V ( G i ) | g ∗ s ( | Y ∗ ∩ V ( G i ) | ) k X i =1 ( | Y | − g ∗ s ( | Y | − η ∗ s · ( | Y | − g ∗ s ( | Y | − | Y | g ∗ s ( | Y | ) . c ∗ -monochromatic component disjoint from Y ∪ P is a c i -monochromaticcomponent for some i ∈ [ k ] , and hence contains at most η ∗ s g ∗ s ( η ) vertices. Therefore c ∗ is an ( η ∗ s , g ∗ s ) -bounded L -coloring of G . This proves the lemma.The following lemma is equivalent to Lemma 6.9 except it applies for restricted list as-signments. The proof is identical, so we omit it. Lemma 6.10. Let G be a subgraph-closed family of graphs. Let β, r be functions with domain N such that β ( x ) > x and r ( x ) ∈ N for every x ∈ N .Assume that for every s ∈ N , there exists η ∈ N and a nondecreasing function g such thatfor every G ∈ G with no K s,t s -subgraph, where t s := max { (cid:0) s (cid:1) + 2 , s + 2 } , for every Y ⊆ V ( G ) with | Y | η , and for every restricted ( β ( s ) , r ( s ) , Y ) -list-assignment L of G , there exists an ( η, g ) -bounded L -coloring of G .Then for every s ∈ N with s > , there exists η ∗ ∈ N and a nondecreasing function g ∗ such that for every graph G ∈ G containing no almost ( -subdivision of K s +1 , forevery Y ⊆ V ( G ) with | Y | η ∗ , and for every restricted faithful ( β ( s − , r ( s − , Y ) -list-assignment L of G , there exists an ( η ∗ , g ∗ ) -bounded L -coloring of G . Theorem 6.11. If s ∈ N with s > , then the following hold:1. For every w ∈ N , there exists η ∈ N and a nondecreasing function g such that for everygraph G of treewidth at most w containing no almost ( -subdivision of K s +1 , everysubset Y of V ( G ) with | Y | η and every faithful ( s − , , Y ) -list-assignment of G ,there exists an ( η, g ) -bounded L -coloring of G .2. For every graph H , there exists η ∈ N and a nondecreasing function g such that forevery graph G with no H -minor and no almost ( -subdivision of K s +1 , every subset Y of V ( G ) with | Y | η and every restricted faithful ( s − , , Y ) -list-assignment L of G , there exists an ( η, g ) -bounded L -coloring of G .3. For every d ∈ N with d > and graph H of maximum degree at most d , there exists η ∈ N and a nondecreasing function g such that for every graph G with no H -subdivisionand no almost ( -subdivision of K s +1 , every subset Y of V ( G ) with | Y | η andevery restricted faithful ( s + 3 d − , , Y ) -list-assignment L of G , there exists an ( η, g ) -bounded L -coloring of G .4. There exists η ∈ N and a nondecreasing function g such that for every graph G withno K s +1 -subdivision, every subset Y of V ( G ) with | Y | η and every restricted faithful (4 s − , , Y ) -list-assignment L of G , there exists an ( η, g ) -bounded L -coloring of G .Proof. Statement 1 follows from Theorem 5.3 and Lemma 6.9 by taking G to be the set ofgraphs of treewidth at most w , β ( s ) = s and r ( s ) = 1 .Statement 2 follows from Theorem 5.10 and Lemma 6.10 by taking G to be the set ofgraphs with no H -minor, β ( s ) = s and r ( s ) = 2 .Statement 3 follows from Lemmas 6.7 and 6.10 by taking G to be the set of graphs withno H -subdivision, β ( s ) = 3 d + s − and r ( s ) = 2 . Note that β ( s ) > s since d > . And d + s > since d > and s > .Statement 4 follows from Statement 3 by taking H = K s +1 .When Y = ∅ , every ( s, r, Y ) -list-assignment is faithful. Thus, Theorem 6.11 implies thatfor all s, d, w ∈ N with s > and d > , for every graph H , there exists η ∈ N such that:143. For every graph G with treewidth at most w and containing no almost ( -subdivisionof K s +1 , and for every list-assignment L of G with | L ( v ) | > s for every v ∈ V ( G ) , thereexists an L -coloring with clustering η (Theorem 1.3 for s > ).2. For every graph G containing no almost ( -subdivision of K s +1 and with no H -minor,there exists an ( s + 1) -coloring of G with clustering η (Theorem 1.4 for s > ).3. If the maximum degree of H is at most d , then for every graph G with no H -subdivisionand no almost ( -subdivision of K s +1 , there exists an ( s + 3 d − -coloring of G withclustering η (Theorem 1.6 for s > and d > ).4. For every graph G with no K s +1 -subdivision, there exists a (4 s − -coloring of G withclustering η (Theorem 1.8 for s > ).Note that when s = 1 , graphs with no K s +1 -subgraph have no edge, so they are 1-colorablewith clustering 1. This together with Lemma 6.6 complete the proof of Theorems 1.3, 1.4,1.6 and 1.8. The Four Color Theorem [2, 60] is best possible, even in the setting of clustered coloring.That is, for all c there are planar graphs for which every 3-coloring has a monochromaticcomponent of size greater than c ; see [78]. These examples have unbounded maximum de-gree. 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r + 1 .)(L4) For every v ∈ V ( G ) − N G [ Y ] , we have | L ( v ) | = s + r .20L5) For every v ∈ V ( G ) − Y , we have | L ( v ) | > r + 1 .We say that an ( s, r, Y ) -list-assignment L of G is restricted if:(L1’) L ( v ) ⊆ [ s + r ] for every v ∈ V ( G ) .We use r = 1 for the bounded treewidth case (Theorem 1.14), r = 2 for excluded minors(Theorem 1.2), and r = s + 1 for excluded odd minors (Theorem 1.16). We use restricted listassignments when dealing with excluded odd minors in Section 5.4. Note that a restricted ( s, , Y ) -list-assignment coincides with an ( s, Y, , -list-assignment in Section 5.3. K s,t -Subgraph The following theorem is the main contribution of this section. s ( Y i − ) ∪ N
s + 1 − | N ( v ) ∩ Y i | .Note that | L i ( u ) | = | L i − ( u ) | = 1 for u ∈ N ( v ) ∩ Y i . Suppose that v ∈ N s ( Y i − ) L i ( u ) and | X v | > | L i − ( v ) | − | N ( v ) ∩ N > s ( Y i − ) | . By (L3), | L i − ( v ) | = s + 1 − | N ( v ) ∩ Y i − | . Thus | X v | = s +1 −| N ( v ) ∩ Y i − | −| N ( v ) ∩ N > s ( Y i − ) | = s +1 −| N ( v ) ∩ Y i | , as desired. Now consider v ∈ N s ( Y i − ) ∪ N s + 1 − | N ( v ) ∩ Y i | , as desired. This shows that L i is well defined.We now show that { v ∈ V ( G ) : | L i ( v ) | = 1 } = Y i . This implies that (L2) and (L5)are satisfied. If v ∈ Y i then | L ( v ) | = 1 by construction. If v ∈ N ( s + 1) − ( s − . Thus | L ( v ) | = 1 if and only if v ∈ Y i , implying { v ∈ V ( G ) : | L i ( v ) | = 1 } = Y i .We now show that L i is an ( s, , Y i ) -list-assignment. For all v ∈ V ( G ) , since | L i ( v ) | | L ( v ) | and L i ( v ) = ∅ , we have | L i ( v ) | ∈ [ s + 1] and (L1) is satisfied. By construction,for all v ∈ N s ( Y i − ) | c | Y i − | by Lemma 3.6. Hence | Y i | = | Y i − | + | N > s ( Y i − ) | ( c + 1) | Y i − | . By induction, | Y i − | ( c + 1) i − | Y | , implying | Y i | ( c + 1) i | Y | , as desired.Let L ′ := L | Y | and Y ′ := Y | Y | . Claim 3.8.1 implies that L ′ is an ( s, , Y ′ ) -list-assignmentfor G , with | Y ′ | | Y | ( c + 1) | Y | .Since N ( Y ) = ∅ , it follows that | Y ′ | > | Y | . By the choice of counterexample, ( G, Y ′ , L ′ ) isnot a counterexample. Thus there is an ( η, g ) -bounded L ′ -coloring of G , which is an L -coloringof G since L ′ ( v ) ⊆ L ( v ) for every vertex v ∈ V ( G ) .It remains to show that the union of the monochromatic components that intersect Y hasat most | Y | g ( | Y | ) vertices. Let M be a monochromatic component that intersects Y . Thus22 j is in M for some j ∈ [1 , | Y | ] . Since L ( x j ) = { α } for some color α , every vertex in M iscolored α . Observe that N ( Y j − ) = N > s ( Y j − ) ∪ N s ( Y j − ) , and thus no vertex in N > s ( Y j − ) is colored α . Since L j − satisfies(L3), for each vertex v ∈ N s ( Y j − ) ∪ N
r + 1 > | F | > | L ( a ) ∩ F | . L ′ ( a ) is chosen so that | L ′ ( a ) ∩ F | is maximum among all subsets of L ( v ) − { L ′ ( w ) : w ∈ N G ( v ) ∩ ( W − Y ) } of size | L ( v ) | − | N G ( v ) ∩ ( W − Y ) | , and { y ∈ W − Y : L ′ ( y ) ∩ F = ∅} = ∅ ,we know L ′ ( a ) must contain L ( a ) ∩ F . Since L ′ ( a ) ⊆ L ( a ) , we have L ′ ( a ) ∩ F ⊆ L ( a ) ∩ F .Hence L ′ ( a ) ∩ F = L ( a ) ∩ F , a contradiction. So Statement 5 holds.To complete the proof it suffices to show that L ′ is an ( s, r, Y ′ ) -list-assignment of G .We first show that L ′ satisfies (L3). Let v ∈ N r + 1 , and L ′ ( v ) is a subset of L ( v ) − { L ′ ( w ) : w ∈ N G ( v ) ∩ Y ′ } , so L ′ ( v ) ∩ L ′ ( u ) = ∅ for every u ∈ N G ( v ) ∩ Y ′ . So we may assume v ∈ N and L ( v ) ∩ L ( u ) = ∅ for every u ∈ N G ( v ) ∩ Y . Hence | L ′ ( v ) | = | L ( v ) | − | N G ( v ) ∩ ( W − Y ) | = s + r − | N G ( v ) ∩ Y | − | N G ( v ) ∩ ( W − Y ) | = s + r − | N G ( v ) ∩ ( W ∪ Y ) | = s + r − | N G ( v ) ∩ Y ′ | . Furthermore, L ′ ( v ) ⊆ L ( v ) − { L ′ ( w ) : w ∈ W − Y } and L ′ ( u ) = L ( u ) for every u ∈ Y , so L ′ ( v ) ∩ L ′ ( u ) = ∅ for every u ∈ N G ( v ) ∩ ( W ∪ Y ) = N G ( v ) ∩ Y ′ . Hence L ′ satisfies (L3).Let x be a vertex in V ( G ) − N G [ Y ′ ] . Since Y ⊆ Y ′ , we have x ∈ V ( G ) − N G [ Y ] . Since L satisfies (L4), we have | L ( x ) | = s + r . Since Y ′ ∪ N G ( Y ′ ) ⊇ Y ′ ∪ N s ( Y ′ ) . Then | L ′ ( z ) | = | L ( z ) | . Since z Y and L satisfies (L2)–(L5), we have | L ′ ( z ) | > r + 1 . Since L ′ satisfies (L3) and (L4), it implies that L ′ satisfies (L5). Since L ′ satisfies (L3)–(L5), L ′ satisfies (L1) and (L2). Therefore L ′ is an ( s, r, Y ′ ) -list-assignment. This section introduces several definitions related to the list-coloring argument used toprove Theorem 1.17 for graphs of bounded layered treewidth. The idea is to ensure that color i does not appear in the lists of vertices in layers V j with j ≡ i (mod s + 2) . Then eachmonochromatic component is contained within at most s + 1 consecutive layers.Let G be a graph and Z ⊆ V ( G ) . A Z -layering V of G is an ordered partition ( V , V , . . . ) of V ( G ) − Z into (possibly empty) sets such that for every edge e of G − Z , there exists i ∈ N such that both ends of e are contained in V i ∪ V i +1 . Note that a layering is equivalent to an ∅ -layering. For a tree-decomposition ( T, X ) of G , with X = ( X t : t ∈ V ( T )) , the V -width of ( T, X ) is max i ∈ N max t ∈ V ( T ) | X t ∩ V i | . Let G be a graph and let Z ⊆ V ( G ) . For every s ∈ N and ℓ ∈ [ s + 2] , an s -segment ofa Z -layering ( V , V , . . . ) of level ℓ is S a + sj = a V j for some (possibly non-positive) integer a with a ≡ ℓ + 1 (mod s + 2 ), where V i = ∅ for every i . When the integer s is clear from thecontext, we write segment instead of s -segment.Let G be a graph and let s ∈ N . Let Z ⊆ V ( G ) and V = ( V , V , . . . ) be a Z -layering of G . A list-assignment L of G is ( s, V ) -compatible if the following conditions hold:28 L ( v ) ⊆ [ s + 2] for every v ∈ V ( G ) . • i L ( v ) for every i ∈ [ s + 2] and v ∈ S ( V j : j ≡ i (mod s + 2)) .Note that there is no condition on L ( v ) for v ∈ Z .We remark that for every i ∈ [ s + 2] , if v ∈ V ( G ) with i ∈ L ( v ) , then either v ∈ Z , or v belongs to a segment of V with level i . This leads to the following easy observation that wefrequently use. Proposition 4.5. s ( Y ) = ∅ . So Y = N G ( Y ) . Since N G ( Y ) = ∅ , there exists a vertex z ∈ Y . Define Y ′ := Y ∪ { z } and define L ′ to be a ( { z } , F ∪ { ℓ } ) -progress of L . SinceStatement 2 does not hold, | Y ′ | = | Y | + 1 k + 1 η . Since Y = N G ( Y ) and by (L3) L ( v ) ∩ L ( u ) = ∅ for every v ∈ Y and u ∈ N G ( v ) ∩ Y , the union of the monochromaticcomponents with respect to any L ′ -coloring of G intersecting Y is contained in G [ Y ] . SinceStatement 3 does not hold, by Lemma 4.4, there exists an ( η, g ) -bounded L ′ -coloring c ′ of G such that for every x ∈ F , the set of vertices colored x is a stable set. In particular, theunion of the c ′ -monochromatic components intersecting Y is contained in G [ Y ] . So c ′ is an ( η, g ) -bounded L -coloring of G , and Statement 1 holds, a contradiction.Hence N > s ( Y ) = ∅ . Denote Y by { y , y , . . . , y | Y | } . For every i ∈ [ | Y | ] , let ℓ i be theunique element of L ( y i ) ; let ℓ | Y | +1 = ℓ . Define L = L and U = Y . For i > , let L i be a ( N > s ( U i − ) , { ℓ i }∪ F ) -progress and define U i := { v ∈ V ( G ) : | L i ( v ) | = 1 } . Define L ∗ := L | Y | +1 and Y ∗ := U | Y | +1 . Claim 5.1.1. | Y ∗ | h | Y | ( | Y | ) .Proof. We shall prove that | U i | h i − ( | Y | ) by induction on i > . When i = 0 , we have | U | = | Y | = h − ( | Y | ) . Now assume that i > and the claim holds for all smaller i . Byinduction and Lemma 3.5, | U i | = | U i − | + | N > s ( U i − ) | | U i − | + f ( | U i − | ) = h ( | U i − | ) h ( h i − ( | Y | ))= h i − ( | Y | ) . The case i = | Y | + 1 proves the claim.Since Statement 2 does not hold, | Y | k . By Claim 5.1.1, | Y ∗ | h k ( k ) η . Recall thatwe proved that N > s ( U ) = ∅ . So | Y ∗ | > | Y | . And by Lemma 4.4, { v ∈ Y : F ∩ L ( v ) = ∅} = { v ∈ Y ∗ : F ∩ L ∗ ( v ) = ∅} . Claim 5.1.2. s ( U | Y | ) , { ℓ } ∪ F ) -progress of L | Y | , by Lemma 4.4, for every x ∈ F ∪ { ℓ } and y ∈ Y ∗ with x ∈ L ∗ ( y ) , we have { v ∈ N G ( y ) − Y ∗ : x ∈ L ∗ ( v ) } = ∅ . Since Statement 3does not hold, by Claim 5.1.2, there exists an ( η, g ) -bounded L ∗ -coloring c ∗ of G such that107or every x ∈ F , the set of vertices colored x is a stable set. So every c ∗ -monochromaticcomponent contains at most η g ( η ) vertices. By Claims 5.1.1 and 5.1.2, the union of the c ∗ -monochromatic components intersecting Y contains at most | Y ∗ | h | Y | ( | Y | ) g ( | Y | ) vertices. Therefore, c ∗ is an ( η, g ) -bounded L -coloring of G such that for every x ∈ F , theset of vertices colored x is a stable set. So Statement 1 holds. This contradiction proves thelemma. The goal of this section is to prove Theorem 1.14 regarding graphs of bounded treewidthand to set-up machinery for the proofs of Theorems 1.2 and 1.16 in subsequent sections.A tangle T in a graph G of order θ ∈ Z is a set of separations of G of order less than θ such that the following hold:(T1) For every separation ( A, B ) of G of order less than θ , either ( A, B ) ∈ T or ( B, A ) ∈ T .(T2) If ( A i , B i ) ∈ T for i ∈ [3] , then A ∪ A ∪ A = G .(T3) If ( A, B ) ∈ T , then V ( A ) = V ( G ) . Lemma 5.2. r + 1 > | F | and ( S v ∈ V ( A ∩ B ) − Y L A ( v )) ∩ F = ∅ ,we have L A ( z ) contains L ( z ) ∩ F . Hence L ( z ) ∩ F ⊆ L A ( z ) ∩ F ⊆ L ( z ) ∩ F , as claimed.For each v ∈ V ( A ) − ( Y A ∪ Z A ) , let L A ( v ) := L ( v ) . Claim 5.2.2. L A is an ( s, r, Y A ) -list-assignment of G A such that • L A ( v ) ⊆ L ( v ) for every v ∈ V ( G A ) . • { v ∈ Y : F ∩ L ( v ) = ∅} ∩ V ( A ) = { v ∈ Y A : F ∩ L A ( v ) = ∅} . • For every v ∈ V ( G A ) − Y A , we have L A ( v ) ∩ F = L ( v ) ∩ F .Proof. Clearly, L A satisfies (L1). By Claim 5.2.1, L A satisfies (L3). Since L A ( v ) = L ( v ) foreach v ∈ V ( A ) − ( Y A ∪ Z A ) ⊇ V ( A ) − N G A [ Y A ] , we have L A satisfies (L4). Since L A satisfies (L3)and (L4), if v ∈ V ( A ) − Y A with | L ( v ) | < r , then v ∈ N G A ( Y A ) − N r since L satisfies (L5), a contradiction. So L A satisfies (L5). Since L A satisfies (L1) and (L3)–(L5), L A satisfies (L2) and hence is an ( s, r, Y A ) -list-assignment.Clearly, no vertex y ∈ Y A − Y satisfies L A ( y ) ∩ F = ∅ , so { v ∈ Y : F ∩ L ( v ) = ∅} ∩ V ( A ) = { v ∈ Y A : F ∩ L A ( v ) = ∅} . Furthermore, if v ∈ Z A , then L A ( v ) ∩ F = L ( v ) ∩ F by Claim 5.2.1;if v ∈ V ( A ) − ( Y A ∪ Z A ) , then L A ( v ) = L ( v ) , so L A ( v ) ∩ F = L ( v ) ∩ F . Hence for every v ∈ V ( A ) − Y A , we have L A ( v ) ∩ F = L ( v ) ∩ F .Similarly, there exists an ( s, r, Y B ) -list-assignment of G B with L A ( v ) = L B ( v ) for every v ∈ V ( A ∩ B ) , such that: • L B ( v ) ⊆ L ( v ) for every v ∈ V ( G B ) . • { v ∈ Y : F ∩ L ( v ) = ∅} ∩ V ( B ) = { v ∈ Y B : F ∩ L B ( v ) = ∅} . • For every v ∈ V ( G B ) − Y B , we have L B ( v ) ∩ F = L ( v ) ∩ F .109ince Statement 2 does not hold, there exists an ( η, g ) -bounded L A -coloring c A of G A suchthat for every x ∈ F , the set of vertices colored x is a stable set in G A , and there exists an ( η, g ) -bounded L B -coloring c B of G B such that for every x ∈ F , the set of vertices colored x is a stable set in G B . By construction, c A ( v ) = c B ( v ) for every v ∈ V ( A ∩ B ) . Define c ( v ) := c A ( v ) of v ∈ V ( A ) and define c ( v ) := c B ( v ) if v ∈ V ( B ) . Clearly, c is an L -coloringsuch that for every x ∈ F , the set of vertices colored x is a stable set in G .Let C be the union of the c -monochromatic components intersecting Y ∪ V ( A ∩ B ) . Then V ( C ) ∩ Y A = ∅ and V ( C ) ∩ Y B = ∅ . By construction, | V ( C ) | | V ( C ) ∩ V ( A ) | + | V ( C ) ∩ V ( B ) | | Y A | g ( | Y A | ) + | Y B | g ( | Y B | ) ( | Y A | + | Y B | ) g ( | Y | ) (( | Y A | + | Y B | ) − | Y A || Y B | ) g ( | Y | ) . Since | Y A | > θ and | Y B | > θ , | Y | θ + 2 θ ( | Y A | + | Y B | ) θ + 2 θ | Y A | · | Y B | | Y B | · | Y A | · | Y A | · | Y B | | Y A | · | Y B | . Therefore, | V ( C ) | (( | Y A | + | Y B | ) − | Y A || Y B | ) g ( | Y | ) (( | Y | + θ ) − | Y | θ − θ ) g ( | Y | ) | Y | g ( | Y | ) . Since Statement 1 does not hold, c is not an ( η, g ) -bounded L -coloring of G . Since | Y | g ( | Y | ) η g ( η ) , there exists a c -monochromatic component M disjoint from Y ∪ V ( A ∩ B ) containing at least η g ( η ) + 1 vertices. However, since M is disjoint from V ( A ∩ B ) , either M is contained in G A or M is contained in G B . So M is a monochromatic component withrespect to c A or c B . Since c A , c B are ( η, g ) -bounded, M contains at most η g ( η ) vertices, acontradiction. This proves the lemma. Theorem 5.3. , let f i : N → N be the function definedby f i ( x ) := x + f i − ( x ) + f ( f i − ( x )) . Define h := f s +2 .Since L is an ( s, r + 2 , Y ) -list-assignment and L ′ is obtained from L by repeatedly taking ( W i , F i ) -progress for some sets W i , F i with | F i | r + 2 and { ℓ } ∪ [ s + 3 , s + 2 + r ] ⊆ F i for all i such that L ′ ( v ) ∩ [ s + 3 , s + 2 + r ] = L ( v ) ∩ [ s + 3 , s + 2 + r ] for every v ∈ V ( G ) − Y ′ , weknow that L ′ is an ( s, r + 2 , Y ′ ) -list-assignment of G satisfiying (R5) such that { v ∈ Y ′ : L ′ ( v ) ∩ ( { ℓ } ∪ [ s + 3 , s + 2 + r ]) = ∅} = { v ∈ Y : L ( v ) ∩ ( { ℓ } ∪ [ s + 3 , s + 2 + r ]) = ∅} by Lemma 4.4. So L ′ satisfies (R1)–(R5), and Statement 2 holds. Hence Statements 1 and 2hold.For every i , let U i , Y ( i )1 , L ( i ) be the sets and list-assignment mentioned in the definitionof a ( Z, ℓ ) -growth. By Lemma 3.5, it is easy to verify that | U i | f ( | Y ( i − | ) and | Y ( i )1 | | Y ( i − | + | U i | f i ( | Y ∪ Z | ) for every i > by induction on i . This proves Statement 3.Let c ′ be an L ′ -coloring of G . Let M i be a c ′ -monochromatic component intersecting Y ∪ Z such that all vertices of M i are colored i for some i ∈ [ s + r + 2] . If i ∈ { ℓ } ∪ [ s + 3 , s + 2 + r ] ,then V ( M ) ∩ Z − Y = ∅ and V ( M ) ⊆ Y since c ′ is an L -coloring and L satisfies (R3).So we may assume that i ∈ [ s + 2] − { ℓ } . Since L ( i ) is a ( U i , { ℓ, i } ∪ [ s + 3 , s + 2 + r ]) -progress of L ( i − , we have V ( M i ) ∩ U i = ∅ . Since M i is connected and V ( M i ) ∩ U i = ∅ and V ( M i ) ∩ ( Y ∪ Z ) = ∅ , we have either V ( M i ) ⊆ Y ( i − or there exists xy ∈ E ( M i ) suchthat y ∈ Y ( i − and x ∈ N G ( y ) ∩ N , since ( A, B ) is a separation, U i ∩ V ( A ) ⊆ V ( A ∩ B ) ∪ { v ∈ V ( A ) − Y ( i − : | N G ( v ) ∩ Y ( i − ∩ V ( A ) | > s } . Since Z ⊆ V ( A ∩ B ) , by Lemma 3.5, it is easy to prove by induction on i that | U i ∩ V ( A ) | | V ( A ∩ B ) | + f ( | Y ( i − ∩ V ( A ) | ) and | Y ( i )1 ∩ V ( A ) | | Y ( i − ∩ V ( A ) | + | U i ∩ V ( A ) | f i ( | V ( A ∩ B ) | + | Y ∩ V ( A ) | ) . Statement 5 holds by taking i = s + 2 .113 emma 5.8. For all s, t, t ′ ∈ N , there exist θ ∗ ∈ N and nondecreasing functions g ∗ , η ∗ withdomain N such that if G is a graph with no K s,t -subgraph, θ ∈ N with θ > θ ∗ , η ∈ N with η > η ∗ ( θ ) , Y ⊆ V ( G ) with θ < | Y | η , ℓ ∈ [0 , s + 2] , r ∈ N , L is an ( s, Y , ℓ, r ) -list-assignment of G , g is a nondecreasing function with domain N and with g > g ∗ , and T is atangle in G of order θ that does not control a K t ′ -minor, where T = { ( A, B ) : | V ( A ∩ B ) | <θ, | V ( A ) ∩ Y | θ } , then either:1. there exists an ( η, g ) -bounded L -coloring of G such that for every x ∈ [ s + 3 , s + 2 + r ] ,the set of vertices colored x is a stable set, or2. there exist ( A ∗ , B ∗ ) ∈ T , a set Y A ∗ with | Y A ∗ | η ∗ ( θ ) and Y ∩ V ( A ∗ ) ⊆ Y A ∗ ⊆ V ( A ∗ ) ,and an ( s, Y A ∗ , ℓ, r ) -list-assignment L A ∗ of G [ V ( A ∗ )] such that there exists no ( η, g ) -bounded L A ∗ -coloring of G [ V ( A ∗ )] such that for every x ∈ [ s + 3 , s + 2 + r ] , the set ofvertices colored x is a stable set.Proof. Define the following: • Let f be the function f s,t in Lemma 3.5. • Let f : N → N be the identity function. For every i ∈ N , let f i : N → N be thefunction defined by f i ( x ) := f i − ( x ) + f ( f i − ( x )) . • Let κ , ρ , ξ , θ be the integers κ, ρ, ξ, θ in Theorem 5.6 taking H = K t ′ . • Let h : N → N be the function in Lemma 5.7 taking s = s and t = t . • Let θ ∗ := θ + 2 ρ + ξ + 3 . • Let η ∗ : N → N be the function defined by η ∗ ( x ) := h (4 x )+ f s +3 ( h (4 x )+ x )+ f ( h (4 x )+ f s +3 ( h (4 x ) + x )) for every x ∈ N . • Let σ be the maximum Euler genus of a surface in which K t ′ cannot be embedded. • For every x ∈ N , let w ( x ) := (2 σ + 3)(2 ρ + 1) · f s +3 ( h (4 x ) + x ) . • For every x ∈ N , let η ( x ) be the number η ∗ in Lemma 4.13 taking s = s , t = t , w = w ( x ) , k = h ( x + ξ ) and ξ = h ( x + ξ ) . • Let g ∗ : N → N be the function defined by g ∗ ( x ) := η (2 x )+ h ( x + ξ ) for every x ∈ N .Let G be a graph with no K s,t -subgraph, θ ∈ N with θ > θ ∗ , η ∈ N with η > η ∗ ( θ ) , Y a subset of V ( G ) with | Y | η , ℓ ∈ [0 , s + 2] , r ∈ N , g a function with g > g ∗ , and L an ( s, Y , ℓ, r ) -list-assignment of G . Let T = { ( A, B ) : | V ( A ∩ B ) | < θ, | V ( A ) ∩ Y | θ } .Assume that T is a tangle in G of order θ that does not control a K t ′ -minor.Suppose that there exists no ( η, g ) -bounded L -coloring of G such that for every x ∈ [ s + 3 , s + 2 + r ] , the set of vertices colored x is a stable set, and suppose that for everyseparation ( A, B ) ∈ T , every set Y A with | Y A | η ∗ ( θ ) and Y ∩ V ( A ) ⊆ Y A ⊆ V ( A ) andevery ( s, Y A , ℓ, r ) -list-assignment L A of G [ V ( A )] , there exists an ( η, g ) -bounded L A -coloringof G [ V ( A )] such that for every x ∈ [ s + 3 , s + 2 + r ] , the set of vertices colored x is a stableset.Since T does not control a K t ′ -minor, by Theorem 5.6, there exist Z ⊆ V ( G ) with | Z | ξ ,a ( T − Z ) -central segregation S of G − Z of type ( κ , ρ ) , and a proper arrangement of S in a surface Σ in which K t ′ cannot be embedded. Let S := { ( S, Ω) ∈ S : | Ω | } , and let S := S − S . Since S is of type ( κ , ρ ) , |S | κ and every member of S is a ρ -vortex.For each ( S, Ω) ∈ S , let ( A S , B S ) be the separation of G such that S ⊆ A S , V ( A S ) = V ( S ) ∪ Z , V ( A S ∩ B S ) = Ω ∪ Z , and subject to these conditions, | E ( A S ) | is minimal. Let114 := { ( A S , B S ) : ( S, Ω) ∈ S } . By Theorem 5.5, for each ( S, Ω) ∈ S , there exists a vorticaldecomposition ( P S , X S ) of ( S, Ω) of adhesion at most ρ . For each ( S, Ω) ∈ S and each bag X of ( P S , X S ) , let ∂X := ( X ∩ Ω) ∪ { v ∈ X : v belongs to a bag of ( P S , X S ) other than X } . Let ( A S,X , B | L ′ ( y ) | − r + | N G ( y ) ∩ Y ′ | − | N G ( y ) ∩ Y ′ ∩ V ( G ′′ ) | − ( s + r + 2 − | N G ( y ) ∩ Y ′ | ) − r + | N G ( y ) ∩ Y ′ | − | N G ( y ) ∩ Y ′ ∩ V ( G ′′ ) | − s + 1 − | N G ( y ) ∩ Y ′ ∩ V ( G ′′ ) | = s + 1 − | N G ′′ ( y ) ∩ Y ′ | and L ′L ( y ) ∩ L ′L ( u ) = ∅ for every u ∈ N G ( y ) ∩ Y ′ ∩ V ( G ′′ ) . • For every v ∈ V ( G ′′ ) − ( Y ′ ∪ N G ′′ ( Y ′ )) , | L L ( v ) | > | L ′ ( v ) | − r + | N G ( v ) ∩ Y ′ | − | N G ( v ) ∩ Y ′ ∩ V ( G ′′ ) | − > s + 1 . • For every v ∈ V ( G ′′ ) − Y ′ , | L L ( v ) | > | L ′ ( v ) | − r + | N G ( v ) ∩ Y ′ | − | N G ( v ) ∩ Y ′ ∩ V ( G ′′ ) | − > . Hence there exists L L ,G ′′ ( v ) ⊆ L L ( v ) for every v ∈ V ( G ′′ ) such that L L ,G ′′ is an ( s, , Y ′ ∩ V ( G ′′ )) -list-assignment of G ′′ . Therefore, ( Y ′ ∩ V ( G ′′ ) , L L ,G ′′ ) is a V -standard pair for G ′′ .By Lemma 5.7, | Y ′ | h ( | Y | + | Z | ) h ( η + ξ ) . So by Lemma 4.13, there exists an L L ,G ′′ -coloring c L of G ′′ with clustering η ( η + θ ) . By further coloring each vertex y in Y ′ − V ( G ′′ ) with the unique element in L ′ ( y ) , we extend the coloring c L to be an L ′ -coloring of G ′ withclustering η ( η + θ ) , and { v ∈ V ( G ′ ) : c L ( v ) = x } = { v ∈ Y ′ : L ′ ( v ) = { x }} is a stable set forevery x ∈ [ s + 3 , s + 2 + r ] . Claim 5.8.2.
s + 3 , so L ′ ( v ) ∩ [ s + 2] = ∅ 6 = L ′ ( v ) ∩ [ s + 3 , s + 1] , and since | N G ( v ) ∩ Y ′′ − Y ′ | , we know that L ′′ ( v ) can be chosensuch that | ( L ′ ( v ) − L ′′ ( v )) ∩ [ s + 3 , s + 1] | = |{ y ∈ N G ( v ) ∩ Y ′′ − Y ′ : L ′′ ( y ) ∈ [ s + 3 , s + 1] }| .Therefore, for every v ∈ N s } , { ℓ } ∪ [ s + 3 , s + 1]) -progress of L ′′ . Let Y ′′′ := { v ∈ V ( G ) : | L ′′′ ( v ) | = 1 } . By Lemma 4.4 and Claim 5.10.8, L ′′′ is an ( s, s + 1 , Y ′′′ ) -list-assignment of G and satisfies (R3)-(R5). Hence L ′′′ is an ( s, Y ′′′ , ℓ, s − -list-assignment. Furthermore, for every ( A, B ) ∈ L , since V ( A ∩ B ) ⊆ Y ′′ , we have | Y ′′′ ∩ V ( A ) | | Y ′′ ∩ V ( A ) | + f ( | Y ′′ ∩ V ( A ) | ) by Lemma 3.5, so | Y ′′′ ∩ V ( A ) | h (4 θ ′ ) + 1 + f ( h (4 θ ′ ) + 1) η by Claim 5.10.8.Let L ∗ be an ( ∅ , ℓ ) -growth of L ′′′ , and let Y ∗ := { v ∈ V ( G ) : | L ∗ ( v ) | = 1 } . By Lemma 5.7, L ∗ is an ( s, Y ∗ , ℓ, s − -list-assignment of G such that: • For every L ∗ -coloring of G , every monochromatic component intersecting Y ′′′ is con-tained in G [ Y ∗ ] . • For every ( A, B ) ∈ L , we have | Y ∗ ∩ V ( A ) | h ( | A ∩ B | + | Y ′′′ ∩ V ( A ) | ) h ( θ ′ + h (4 θ ′ ) +1 + f ( h (4 θ ′ ) + 1)) η η .For every ( A, B ) ∈ L , since V ( A ∩ B ) ⊆ Y ∗ , we have L ∗ | G [ V ( A )] is an ( s, Y ∗ , ℓ, s − -list-assignment of G [ V ( A )] such that |{ y ∈ V ( A ) : | L ∗ ( y ) | = 1 }| = | Y ∗ ∩ V ( A ) | η η .For every ( A, B ) ∈ L , since ( A, B ) ∈ T , we have | V ( A ) | < | V ( G ) | . Therefore, for every ( A, B ) ∈ L , by the minimality of G , there exists an ( η, g ) -bounded L ∗ | G [ V ( A )] -coloring c ∗ A of G [ V ( A )] such that { v ∈ V ( A ) : c ∗ A ( v ) = x } is a stable set in G [ V ( A )] for every x ∈ [ s +3 , s +1] .Since T ( A,B ) ∈L V ( B ) ⊆ V ( G ′ ) ⊆ Y ∗ and L is a location, for every v ∈ V ( G ) − Y ∗ , thereuniquely exists ( A v , B v ) ∈ T such that v ∈ V ( A v ) − V ( B v ) . Let c ∗ be the following function: • For every v ∈ Y ∗ , let c ∗ ( v ) be the unique element in L ∗ ( v ) . • For every v ∈ V ( G ) − Y ∗ , let c ∗ ( v ) := c ∗ A v ( v ) .Clearly, c ∗ is well-defined and is an L ∗ -coloring of G .Suppose that there exists x ∗ ∈ [ s + 3 , s + 1] such that the set { v ∈ V ( G ) : c ∗ ( v ) = x ∗ } isnot a stable set in G . Then there exists an edge e of G such that both ends of e are colored x ∗ under c ∗ . If there exists ( A, B ) ∈ L such that e belongs to A , then { v ∈ V ( A ) : c ∗ A ( v ) = x ∗ } isnot a stable set in G [ V ( A )] , a contradiction. So e ∈ T ( A,B ) ∈L B ⊆ G ′ , and hence { v ∈ V ( G ′ ) : c ′ ( v ) = x ∗ } is not a stable set in G ′ , contradicting Claim 5.10.7.This shows that for every x ∈ [ s + 3 , s + 1] , the set { v ∈ V ( G ) : c ∗ ( v ) = x } is a stable setin G . Hence c ∗ is not ( η, g ) -bounded.Since L ∗ ( v ) ⊆ L ′ ( v ) for every v ∈ V ( G ) , c ∗ is an L ′ -coloring of G . Since L ′ is an ( Z, ℓ ) -growth of L , by Lemma 5.7, every c ∗ -monochromatic component intersecting Y ∪ Z is con-tained in G [ Y ′ ] . So the union of c ∗ -monochromatic components intersecting Y ∪ Z is containedin G [ Y ′ ] and hence contains at most | Y ′ | h ( η + ξ ) η g (0) g ( | Y | ) g ( η ) . In particu-lar, the union of c ∗ -monochromatic components intersecting Y contains at most | Y | g ( | Y | ) vertices.Hence there exists a c ∗ -monochromatic component M disjoint from Y ∪ Z such that M contains more than η g ( η ) vertices. Suppose that V ( M ) ∩ V ( G ′ ) = ∅ . Since V ( M ) ∩ Z = ∅ and | V ( A ∩ B ) − Z | for every ( A, B ) ∈ L , G [ V ( M ) ∩ V ( G ′ )] is connected. Since c ∗ ( v ) = c ′ ( v ) for every v ∈ V ( G ′ ) , M ∩ G ′ is a c ′ -monochromatic component, so | V ( M ∩ G ′ ) | η byClaim 5.10.7. For every ( A, B ) ∈ L , if V ( M ) − V ( B ) = ∅ , then V ( M ) ∩ V ( A ∩ B ) − Z = ∅ since V ( M ) is disjoint from Z . By the definition of L , if ( A , B ) , ( A , B ) are distinctmembers of L , then V ( A ∩ B ) − Z = V ( A ∩ B ) − Z . Let L ′ = { ( A, B ) ∈ L : V ( M ) ∩ ( A ∩ B ) − Z = ∅} . So |L ′ | | V ( M ) ∩ V ( G ′ ) | η . For every ( A, B ) ∈ L ′ , since V ( M ) intersects V ( A ∩ B ) − Z ⊆ Y ′′′ ∩ V ( A ) , G [ V ( M ) ∩ V ( A )] is a c ∗ A -monochromatic componentintersecting Y ′′′ ∩ V ( A ) ⊆ Y ∗ ∩ V ( A ) , so by Lemma 5.7, G [ V ( M ) ∩ V ( A )] is contained in G [ Y ∗ ∩ V ( A )] and hence contains at most | Y ∗ ∩ V ( A ) | η vertices. Therefore, | V ( M ) | | V ( M ) ∩ V ( G ′ ) | · η η η g (0) η g ( η ) , a contradiction.Hence V ( M ) ∩ V ( G ′ ) = ∅ . In particular, V ( M ) − V ( B ) = ∅ for some ( A, B ) ∈ L . For every ( A ′ , B ′ ) ∈ L , since V ( A ′ ∩ B ′ ) ⊆ T ( A,B ) ∈L V ( B ) ⊆ V ( G ′ ) , V ( M ) is disjoint from V ( A ′ ∩ B ′ ) .So if there exist ( A , B ) , ( A , B ) ∈ L such that V ( M ) − V ( B ) = ∅ 6 = V ( M ) − V ( B ) , thensince L is a location and M is connected, V ( M ) intersects V ( A ∩ B ) , a contradiction. Sothere uniquely exists ( A ∗ , B ∗ ) ∈ L such that V ( M ) − V ( B ∗ ) = ∅ . Since V ( M ) is disjoint from V ( A ∗ ∩ B ∗ ) , we have V ( M ) ⊆ V ( A ∗ ) − V ( B ∗ ) . Hence M is a c ∗ A ∗ -monochromatic component.Since c ∗ A ∗ is an ( η, g ) -bounded L ∗ | G [ V ( A ∗ )] -coloring, M contains at most η g ( η ) vertices, acontradiction. This proves the lemma. This section proves Theorem 1.15. The proof depends on the following result of Kang andOum [37]. For graphs G and H , let G + H be the graph obtained from the disjoint union of G and H by adding all edges with one end in V ( G ) and one end in V ( H ) . Let K ∗ s,t := K s + I t ,where I t is the graph on t vertices with no edges. Lemma 5.11 (Corollary of [37, Lemma 5.1]) . Let t, d ∈ N with d > . If there exists apositive integer η > t − such that every odd K t +1 -minor-free graph containing no bipartite K ∗ t,t +1 -subdivision has a d -coloring with clustering λ , then every odd K t +1 -minor-free graphhas a ( d + 4 t − -coloring with clustering λ .Proof. Let F be the set of all graphs whose every component contains at most λ vertices. So F is a class of graphs closed under isomorphism and taking disjoint union, and F containsall graphs on at most t − vertices. Since every graph with no odd K t +1 -minor and nobipartite K ∗ t,t +1 -subdivision has a d -coloring with clustering λ , by [37, Lemma 5.1], everyodd K t +1 -minor-free graph has a ( d + 4 t − -coloring with clustering λ .We now prove Theorem 1.15. Theorem 5.12.