(4,2)-choosability of planar graphs with forbidden structures
Zhanar Berikkyzy, Christopher Cox, Michael Dairyko, Kirsten Hogenson, Mohit Kumbhat, Bernard Lidický, Kacy Messerschmidt, Kevin Moss, Kathleen Nowak, Kevin F. Palmowski, Derrick Stolee
aa r X i v : . [ m a t h . C O ] D ec (4 , Zhanar Berikkyzy Christopher Cox Michael Dairyko Kirsten Hogenson Mohit Kumbhat Bernard Lidick´y , Kacy Messerschmidt Kevin Moss Kathleen Nowak Kevin F. Palmowski Derrick Stolee , September 18, 2018
Abstract
All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored usinglists of size four has received significant attention. In terms of constraining the structure of thegraph, for any ℓ ∈ { , , , , } , a planar graph is 4-choosable if it is ℓ -cycle-free. In terms ofconstraining the list assignment, one refinement of k -choosability is choosability with separation .A graph is ( k, s ) -choosable if the graph is colorable from lists of size k where adjacent verticeshave at most s common colors in their lists. Every planar graph is (4 , , , chorded ℓ -cycle is an ℓ -cycle with one additional edge. Wedemonstrate for each ℓ ∈ { , , } that a planar graph is (4 , ℓ -cycles. A proper coloring is an assignment of colors to the vertices of a graph G such that adjacent verticesare assigned distinct colors. A ( k, s ) -list assignment L is a function that assigns a list L ( v ) of k colors to each vertex v so that | L ( v ) ∩ L ( u ) | ≤ s whenever uv ∈ E ( G ). A proper coloring φ of G suchthat φ ( v ) ∈ L ( v ) for all v ∈ V ( G ) is called an L -coloring . We say that a graph G is ( k, s ) -choosable if, for any ( k, s )-list assignment L , there exists an L -coloring of G . We call this variation of graphcoloring choosability with separation. Note that when a graph is ( k, k )-choosable, we simply sayit is k -choosable . Observe that if G is ( k, t )-choosable, then G is ( k, s )-choosable for all s ≤ t. Anotable result from Thomassen [11] states that every planar graph is 5-choosable, so it follows thatall planar graphs are (5 , s )-choosable for all s ≤ . Forbidding certain structures within a planar graph is a common restriction used in graphcoloring. Theorem 1.2 summarizes the current knowledge on (3 , , Department of Mathematics, Iowa State University, Ames, IA, U.S.A. { zhanarb,mdairyko,kahogens,mkumbhat,lidicky,kacymess,kmoss,knowak,kpalmow,dstolee } @iastate.edu Department of Mathematical Sciences, Carnegie Mellon University, Pittsburg, PA, U.S.A. [email protected] Supported by NSF grant DMS-1266016. Department of Computer Science, Iowa State University, Ames, IA, U.S.A. onjecture 1.1 (ˇSkrekovski [13]) . If G is a planar graph, then G is (3 , -choosable. Theorem 1.2.
A planar graph G is (3 , -choosable if G avoids any of the following structures:- 3-cycles (Kratochv´ıl, Tuza, Voigt [9]).- 4-cycles (Choi, Lidick´y, Stolee [4]).- 5-cycles and 6-cycles (Choi, Lidick´y, Stolee [4]). In this paper, we focus on 4-choosability with separation. Kratochv´ıl, Tuza, and Voigt [9] provedthat all planar graphs are (4 , , , Conjecture 1.3 (Kratochv´ıl, et al. [9]) . If G is a planar graph, then G is (4 , -choosable. Theorem 1.4 (Kratochv´ıl, et al. [9]) . If G is a planar graph, then G is (4 , -choosable. Theorem 1.4 was strengthened by Kierstead and Lidick´y [8], where it is shown that we canallow an independent set of vertices to have lists of size 3 rather than 4.
Theorem 1.5 (Kierstead and Lidick´y [8]) . Let G be a planar graph and I ⊆ V ( G ) be an independentset. If L assigns lists of colors to V ( G ) such that | L ( v ) | ≥ for every v ∈ I , and | L ( v ) | = 4 forevery v ∈ V ( G ) \ I , and | L ( u ) ∩ L ( v ) | ≤ for all uv ∈ E ( G ) , then G has an L -coloring. In addition to the work summarized above, there are several results regarding 4-choosability.A graph is k -degenerate if each of its subgraphs has a vertex of degree at most k . Euler’s formulaimplies a planar graph with no 3-cycles is 3-degenerate and hence 4-choosable. This and othersimilar results are listed below in Theorem 1.6. For the last result in Theorem 1.6, note that a chorded ℓ -cycle is an ℓ -cycle with an additional edge connecting two of its non-consecutive vertices. Theorem 1.6.
A planar graph G is 4-choosable if G avoids any of the following structures:- 3-cycles (folklore).- 4-cycles (Lam, Xu, Liu, [10]).- 5-cycles (Wang and Lih [14]).- 6-cycles (Fijavz, Juvan, Mohar, and ˇSkrekovski [7]).- 7-cycles (Farzad [6]).- Chorded 4-cycles and chorded 5-cycles (Borodin and Ivanova [3]). Our main results in this paper are listed below in Theorem 1.7. Note that a doubly-chorded ℓ -cycle is a chorded ℓ -cycle with an additional edge. Theorem 1.7.
A planar graph G is (4 , -choosable if G avoids any of the following structures:- Chorded 5-cycles.- Chorded 6-cycles.- Chorded 7-cycles.- Doubly-chorded 6-cycles and doubly-chorded 7-cycles. We prove each case of Theorem 1.7 separately. In Section 4, we forbid chorded 5-cycles (seeTheorem 4.1). In Section 5, we forbid chorded 6-cycles (see Theorem 5.1); we use parts of thisproof to also prove the case when forbidding doubly-chorded 6-cycles and doubly-chorded 7-cycles(see Corollary 5.2). In Section 6, we forbid chorded 7-cycles (see Theorem 6.2). There are manyfeatures common to all of these proofs, which we detail in Sections 2 and 3.2 .1 Preliminaries and Notation
Refer to [15] for standard graph theory terminology and notation. Let G be a graph with a vertexset V ( G ) and an edge set E ( G ); let n ( G ) = | V ( G ) | . We use K n , C n , and P n to denote the completegraph, cycle graph, and path graph, respectively, each on n vertices. The open neighborhood of avertex, denoted N ( v ), is the set of vertices adjacent to v in G ; the closed neighborhood , denoted N [ v ], is the set N ( v ) ∪ { v } . The degree of a vertex v , denoted d G ( v ), is the number of verticesadjacent to v in G ; we write d ( v ) when the graph G is clear from the context. If the degree of avertex v is k , we call v a k -vertex ; if the degree of v is at least k , we call v a k + -vertex . The length of a face f , denoted ℓ ( f ), is the length of the face boundary walk. If the length of a face f is k , wecall f a k -face ; if the length of f is at least k , we call f a k + -face . All of our main results use the discharging method. We refer the reader to the surveys by Borodin [2]and Cranston and West [5] for an introduction to discharging, which is a method commonly usedto obtain results on planar graphs. For real numbers a v , a f , b , we define initial charge values µ ( v ) = a v d ( v ) − b for every vertex v and ν ( f ) = a f ℓ ( f ) − b for every face f . If a v > a f > a v + 2 a f = b >
0, then Euler’s formula implies that P v µ ( v ) + P f ν ( f ) = − b , and the totalcharge on the entire graph is negative. We then define discharging rules that describe a methodfor moving charge value among vertices and faces while conserving the total charge value. Wedemonstrate that if G is a “minimal counterexample” to our theorem, then every vertex and faceends with nonnegative charge after the discharging process, which is a contradiction. Intuitively,this process works well when forbidding a structure (such as a short chorded cycle) with low charge.In Section 3, we concretely define reducible configurations . Loosely, a reducible configuration isa structure C in a graph G with (4 , L where any L -coloring of G − C extends to an L -coloring of G . If we are looking for a minimal example of a graph that is not (4 , + -faces and 4 + -vertices to 3-faces. However, as we forbid chorded 6-cycles orchorded 7-cycles, there may be many 3-faces very close to each other.If G is a plane graph and G ∗ is its dual, then let F be the set of 3-faces of G and let G ∗ bethe induced subgraph of G ∗ with vertex set F . A cluster is a maximal set of 3-faces that areconnected in G ∗ , i.e., a connected component of G ∗ . Note that two 3-faces sharing an edge areadjacent in G ∗ , and two 3-faces sharing only a vertex are not adjacent in G ∗ . See Figure 1 for a listof the clusters with maximum cycle length six and every internal vertex of degree at least four. Inthese figures, the outer cycle is not necessarily a facial cycle, any area filled with gray is not a face,and a pair of square vertices represent a single vertex. Additionally, bold edges describe separating vf f f f f v u u u u f f f f (K3) (K4) (K5a) (K5b) v u u u u vf f f f f f f f v u u u u u f f f f u u u u u u f f f f (K5c) (K6a) (K6b) (K6c) v u u u u wf f f f g v u u u u u f f f f f u zw u f f (K6d) (K6e) (K6f) (K6g)(K6h) (K6i) (K6j) (K6k)(K6l) (K6m) (K6n) (K6o)(K6p) (K6q) (K6r) These are all of the possible clusters with longest cycle at most six and minimum degree four. Bold edgesdemonstrate separating 3-cycles. Gray regions designate cycles that are not faces. We group our clusters by thelength of the longest cycle in the cluster. Thus a configuration (K n i) has a maximum cycle length of n . Figure 1: Clusters with maximum cycle length at most six.4 -cycles , which are cycles in a plane graph whose exterior and interior regions both contain verticesnot on the cycle. These figures are based on the list of clusters used by Farzad [6] in the proof that7-cycle-free planar graphs are 4-choosable.For k ∈ { , } , there is exactly one way to arrange k triangle is a clustercontaining exactly one 3-face; see (K3). A diamond is a cluster containing exactly two 3-faces; see(K4). For k ≥
3, there are multiple ways to arrange k k -fan is a cluster of k k + 1; see (K5a) and (K6b). A k -wheel is a cluster of k k ; see (K5b) and (K6e).Note that the vertex incident to all faces of a 3-wheel has degree 3. A k -strip is a cluster of k f , . . . , f k where the boundaries of the 3-faces are disjoint except that f i and f i +1 share anedge for i ∈ { , . . . , k − } and f i and f i +2 share a vertex for i ∈ { , . . . , k − } ; see (K5a) and(K6a).If f , . . . , f k are the 3-faces in a cluster, then we will prove that the total charge on f , . . . , f k after discharging is nonnegative. Thus, some of the 3-faces may have negative charge, but this isbalanced by other 3-faces in the cluster having positive charge. Hence, our proofs end with a listof all possible cluster types and verifying that each has nonnegative total charge.While there are 23 total clusters that avoid chorded 7-cycles, we do not have that many cases tocheck. The clusters (K5c) and (K6g)–(K6r) have three bold edges, demonstrating a separating 3-cycle. We avoid checking these cases by using a strengthened coloring statement (see Theorem 6.2)that allows our minimal counterexample to not contain any separating 3-cycles. In this section, we describe structures that cannot appear in a minimal counterexample to Theo-rem 1.7. Let G be a graph, f : V ( G ) → N , and s be a nonnegative integer. A graph is f -choosable if G is L -choosable for every list assignment L where | L ( v ) | ≥ f ( v ). An ( f, s ) -list-assignment is alist assignment L on G such that | L ( v ) | ≥ f ( v ) for all v ∈ V ( G ), | L ( v ) ∩ L ( u ) | ≤ s for all edges uv ∈ E ( G ), and L ( u ) ∩ L ( v ) = ∅ if uv ∈ E ( G ) and f ( u ) = f ( v ) = 1. A graph G is ( f, s ) -choosable if G is L -colorable for every ( f, s )-list-assignment L . Definition 3.1. A configuration is a triple ( C, X, ex) where C is a plane graph, X ⊆ V ( C ), andex : V ( C ) → { , , , ∞} is an external degree function. A graph G contains the configuration( C, X, ex) if C appears as an induced subgraph C ′ of G , and for each vertex v ∈ V ( C ), there areat most ex( v ) edges in G from the copy of v to vertices not in C ′ . For a triple ( C, X, ex), definethe list-size function f : V ( C ) → N as f ( v ) = ( − ex( v ) v ∈ X v / ∈ X .
A configuration (
C, X, ex) is reducible if C is ( f, G with (4 , L contains a copy of a reducible configuration( C, X, ex) and G − X is L -choosable, then G is L -choosable.First, we note that if ( C, X, ex) is a reducible configuration, then any way to add an edgebetween distinct vertices of X and lower their external degree by one results in another reducibleconfiguration. 5C1) (C2) (C3) (C4)(C5) (C6) (C7) (C8)(C9) (C10) (C11) (C12)(C13) (C14) (C15)(C16) (C17) (C18)(C19) (C20) (C21) In these configurations, edges with only one endpoint are external edges. Vertices in X are filled with white. Figure 2: Reducible configurations.6C1) (C2) (C4) (C5) (C10) (C11)(C12) (C13) (C14) (C15) (C16)Figure 3: Alon-Tarsi Orientations.
Lemma 3.2.
Let ( C, X, ex) be a reducible configuration, and suppose that x, y ∈ X are nonadjacentvertices with ex( x ) , ex( y ) ≥ . Let ( C ′ , X ′ , ex ′ ) be the configuration where C ′ = C + xy , X ′ = X ,and ex ′ ( v ) = ( ex( v ) v / ∈ { x, y } ex( v ) − v ∈ { x, y } , . Then the configuration ( C ′ , X ′ , ex ′ ) is reducible.Proof. Let f be the list-size function for C and note that C is ( f, f ′ be the list-size function on the configuration ( C ′ , X ′ , ex ′ ), and let L ′ be an ( f ′ , V ( C ′ ). Note that f ′ ( x ) = f ( x ) + 1 and f ′ ( y ) = f ( y ) + 1. Let S = L ′ ( x ) ∩ L ′ ( y ). If | S | < L ′ ( x ) and L ′ ( y ) to S until | S | = 2. Now let S = { a, b } such that a ∈ L ′ ( x ) and b ∈ L ′ ( y ), and define a list assignment L on C by removing a from L ′ ( x )and removing b from L ′ ( y ). Observe that L is an ( f, L -coloring of C . Since L ( x ) ∩ L ( y ) = ∅ , this proper L -coloring of C is also an L ′ -coloring of C ′ .We will use Lemma 3.2 implicitly by assuming that C [ X ] appears as an induced subgraph inour minimal counterexample G . In this section, we prove that configurations (C1)–(C21) shown in Figure 2 are reducible.
We will use the celebrated Alon-Tarsi Theorem [1] to quickly prove that many of our configura-tions are reducible. In fact, configurations that are demonstrated in this way are reducible for4-choosability, not just (4 , D is an orientation of a graph G if G is the underlying undirected graph of D and D has no 2-cycles; let d + D ( v ) and d − D ( v ) be the out- and in-degree of a vertex v in D . An Euleriansubgraph of a digraph D is a subset S ⊆ E ( D ) such that, for every vertex v ∈ V ( D ), the numberof outgoing edges of v in S is equal to the number of incoming edges of v in S . Let EE ( D ) bethe number of Eulerian subgraphs of even size and EO ( D ) be the number of Eulerian subgraphsof odd size. 7 heorem 3.3 (Alon-Tarsi Theorem [1]) . Let G be a graph and f : V ( G ) → N a function. Supposethat there exists an orientation D of G such that d + D ( v ) ≤ f ( v ) − for every vertex v ∈ V ( G ) and EE ( D ) = EO ( D ) . Then G is f -choosable. We call an orientation an
Alon-Tarsi orientation if it satisfies the hypotheses of Theorem 3.3.For a configuration (
C, X, ex) and the associated list-size function f , it suffices to demonstrate anAlon-Tarsi orientation of C with respect to f . See Figure 3 for a list of Alon-Tarsi orientations ofseveral configurations. Corollary 3.4.
The following configurations have Alon-Tarsi orientations and hence are reducible:(C1), (C2), (C4), (C5), (C10), (C11), (C12), (C13), (C14), (C15), (C16).
In the proofs below, we consider a configuration (
C, X, ex) with list-size function f and assumethat an ( f, L is given for C . We will demonstrate that each C is L -colorable.Refer to Figure 2 for drawings of the configurations.First recall the following fact about list-coloring odd cycles. Fact 3.5. If L is a 2-list assignment of an odd cycle, then there does not exist an L -coloring of thecycle if and only if all of the lists are identical. Lemma 3.6. (C3) is a reducible configuration.Proof.
Let v , . . . , v be the vertices of a 4-cycle with chord v v and let v and v have externaldegree 1; the colors c ( v ) and c ( v ) are fixed. Each of v and v have at least one color in theirlists other than c ( v ) and c ( v ). Since | L ( v i ) | ≥ i ∈ { , } , either one of these verticeshas at least two colors available, or L ( v ) ∩ L ( v ) = { c ( v ) , c ( v ) } . In either case, we can extendthe coloring.For the configurations (C6), (C7), and (C8), label the vertices as in Figure 4: label the centervertex v and the outer vertices v , . . . , v , starting with the vertex directly above v , movingclockwise. v v v v v v v v v v v v v v v v v v (C6) (C7) (C8)Figure 4: Vertex labels for configurations (C6), (C7), and (C8). Lemma 3.7. (C6) is a reducible configuration. roof. The colors c ( v ) and c ( v ) are determined. If c ( v ) and c ( v ) are both in L ( v ), then select c ( v ) from L ( v ) \ ( L ( v ) ∪ { c ( v ) , c ( v ) } ); otherwise, select c ( v ) ∈ L ( v ) \ { c ( v ) , c ( v ) } arbitrarily.Define L ′ ( v ) = L ( v ) \ { c ( v ) , c ( v ) , c ( v ) } , L ′ ( v ) = L ( v ) \ { c ( v ) } , and L ′ ( v ) = L ( v ) \ { c ( v ) } and note that | L ′ ( v i ) | ≥ i ∈ { , , } . If | L ′ ( v ) | = | L ′ ( v ) | = 2, then L ′ ( v ) = L ′ ( v ), sothe 3-cycle v v v has an L ′ -coloring by Fact 3.5. Lemma 3.8. (C7) is a reducible configuration.Proof.
If there exists a color a ∈ L ( v ) ∩ L ( v ), start by assigning c ( v ) = c ( v ) = a ; then greedilycolor the remaining vertices in the following order: v , v , v , v . Otherwise, L ( v ) ∩ L ( v ) = ∅ .Suppose that L ( v ) ∩ L ( v ) = ∅ . Select a color c ( v ) ∈ L ( v ). Considering v as an externalvertex and ignoring the edge v v , the 4-cycle v v v v forms a copy of (C4), which is reducibleby Corollary 3.4. Thus, there exists an L -coloring of v , . . . , v ; this coloring extends to v since L ( v ) ∩ L ( v ) = ∅ . If L ( v ) ∩ L ( v ) = ∅ , then there exists an L -coloring by a symmetric argument.Otherwise, there exist colors a ∈ L ( v ) \ L ( v ) and b ∈ L ( v ) \ L ( v ); assign c ( v ) = a and c ( v ) = b . Select c ( v ) ∈ L ( v ) \ { c ( v ) } . Define L ′ ( v ) = L ( v ) \ { c ( v ) , c ( v ) , c ( v ) } and L ′ ( v ) = L ( v ) \ { c ( v ) , c ( v ) } . Note that if | L ′ ( v ) | = | L ′ ( v ) | = 1, then L ( v ) ∩ L ( v ) = { c ( v ) , c ( v ) } andhence L ′ ( v ) ∩ L ′ ( v ) = ∅ . Thus, the coloring extends by greedily coloring v , v , and v . Lemma 3.9. (C8) is a reducible configuration.Proof. If L ( v ) ∩ L ( v ) = ∅ , then greedily color v and v ; what remains is (C4) and the coloringextends. A similar argument works if L ( v ) ∩ L ( v ) = ∅ .If L ( v ) ∩ L ( v ) = ∅ , then | L ( v ) ∩ L ( v ) | = | L ( v ) ∩ L ( v ) | = 1. Select c ( v ) ∈ L ( v ) \ L ( v ), c ( v ) ∈ L ( v ) \ L ( v ). Define L ′ ( v ) = L ( v ) \ { c ( v ) , c ( v ) } , L ′ ( v ) = L ( v ) \ { c ( v ) } , and L ′ ( v ) = L ( v ) \ { c ( v ) } . Observe that we can L ′ -color the 3-cycle v v v by Fact 3.5 and then select c ( v ) ∈ L ( v ) \ { c ( v ) } .If there exists a color a ∈ L ( v ) ∩ L ( v ), start by assigning c ( v ) = c ( v ) = a and then assign c ( v ) ∈ L ( v ) \{ a } . Define L ′ ( v ) = L ( v ) \{ a, c ( v ) } , L ′ ( v ) = L ( v ) \{ a } , and L ′ ( v ) = L ( v ) \{ a } .Observe that the 3-cycle v v v has an L ′ -coloring by Fact 3.5. Lemma 3.10. (C9) is a reducible configuration.Proof.
Consider the vertex v of arbitrary external degree and let c ( v ) be the color assigned to v .Let u and u be the two neighbors of v in the configuration. If we remove c ( v ) from the lists on u and u , observe that at least two colors remain in every list for every vertex of the 5-cycle. Ifthere is no L -coloring of the configuration, then Fact 3.5 asserts that all lists have size two andcontain the same colors; however, this implies that L ( u ) = L ( u ) and | L ( u ) ∩ L ( u ) | = 3, acontradiction. The configurations (C17)–(C21) are special cases of general constructions called template construc-tions .Let (
C, X, ex) be a configuration with vertices u, v ∈ X . A uv -path P is called a special uv -path if all internal vertices of P have degree two in C and external degree two. A uv -path P is calledan extra-special uv -path if all internal vertices of P have external degree two and degree two in C ,except for a consecutive pair xy where ex( x ) = ex( y ) = 1, d ( x ) = d ( y ) = 3, and there is a vertex9 / ∈ X such that z is a common neighbor to x and y , and z is not adjacent to any other verticesin C . Using these special and extra-special paths, we can describe several configurations by thefollowing templates (see Figure 5), consisting of • (B1) a triangle uvw, where ex( u ) = ex( w ) = 2, ex( v ) = 0, an extra-special uv -path P , and aspecial vw -path P , and • (B2) a triangle vwr , where ex( r ) = ∞ , ex( w ) = 1, ex( v ) = 0, a vertex u adjacent to v whereex( u ) = 2, an extra-special uv -path P , and a special vw -path P . uP x yz v P w yxP uz vr w P (B1) (B2) Dotted lines indicate special paths or extra-special paths. Vertices in X are filled with white. Figure 5: Templates for reducible configurations.We make some basic observations about special and extra-special paths that will be used toprove that these templates correspond to reducible configurations.Let P be a special uv -path or an extra-special uv -path. For every color a ∈ L ( u ), let g uP ( a ) bethe set containing each color b ∈ L ( v ) such that assigning c ( u ) = a and c ( v ) = b does not extendto an L -coloring of P . Since we can greedily color P starting at u until reaching v , there is at mostone color in g uP ( a ). Further, g uP ( a ) = ∅ if and only if this greedy coloring process has exactly onechoice for each vertex in P . Thus, if g uP ( a ) = { b } then also g vP ( b ) = { a } .Since L is an ( f, a , a ∈ L ( u ) such that g uP ( a i ) = ∅ . Moreover, observe that if thereare two distinct colors a , a ∈ L ( u ) such that g uP ( a i ) = ∅ , then both a and a are in every listalong P and hence { a , a } ⊆ L ( v ).If P is an extra-special uv -path with 3-cycle xyz where xy is in the path P , then after a coloris assigned to z (as ex( z ) = ∞ ) either one of x or y has three colors available or | L ( x ) ∩ L ( y ) | ≤ P is an extra-special uv -path, then there is at most one color a ∈ L ( u ) such that g uP ( a ) = ∅ . Lemma 3.11.
All configurations matching the template (B1) are reducible.Proof.
Let (
C, X, ex) be a configuration matching the template (B1) and let L be an ( f, L ( u ) = { a , a } . Since P is an extra-special path, there is at least one i ∈ { , } such that g uP ( a i ) = ∅ . Assign c ( u ) = a i , select c ( w ) ∈ L ( w ) \{ a i } and c ( v ) ∈ L ( v ) \ (cid:0) { c ( u ) , c ( w ) } ∪ g wP ( c ( w )) (cid:1) ;the coloring extends to P and P . 10 orollary 3.12. The configurations (C17), (C18), and (C19) match the template (B1), and hencethey are reducible.
Lemma 3.13.
All configurations matching the template (B2) are reducible.Proof.
Let (
C, X, ex) be a configuration matching the template (B2) and let L be an ( f, c ( r ) be the unique color in the list L ( r ). Let L ( u ) = { a , a } . Since P is anextra-special path, there is at least one i ∈ { , } such that g uP ( a i ) = ∅ . Assign c ( u ) = a i .If c ( r ) / ∈ L ( v ), then select c ( w ) ∈ L ( w ), and L ( v ) ∈ L ( v ) \ (cid:0) { c ( u ) , c ( w ) } ∪ g wP ( c ( w )) (cid:1) ; thecoloring extends to P and P .If c ( r ) ∈ L ( v ), then select c ( w ) ∈ L ( w ) \ L ( v ); observe c ( w ) = c ( r ). There exists a color c ( v ) ∈ L ( v ) \ (cid:0) { c ( r ) , c ( u ) } ∪ g wP ( c ( w )) (cid:1) ; the coloring extends to P and P . Corollary 3.14.
Using Lemma 3.2, the configurations (C20) and (C21) match the template (B2),and hence they are reducible.
In this section we show the case of forbidding chorded 5-cycles from Theorem 1.7.
Theorem 4.1. If G is a plane graph not containing a chorded 5-cycle, then G is (4 , -choosable.Proof. Let G be a counterexample minimizing n ( G ) among all plane graphs avoiding chorded 5-cycles with a (4 , L such that G is not L -choosable. Observe that n ( G ) ≥
4; infact, δ ( G ) ≥
4. Since G is a minimal counterexample, G does not contain any of the reducibleconfigurations (C9)–(C21). If ( C, X, ex) is a reducible configuration, then by Lemma 3.2 C doesnot appear as a subgraph of G where d G ( x ) ≤ d C ( x ) + ex( x ) for all x ∈ V ( C ). Further, theconfigurations (C13)–(C21) are large enough that we must consider configurations that are formedby identifying certain pairs of vertices in these configurations. In Appendix A, we concretely checkall vertex pairs that avoid creating a chorded 5-cycle and find that all resulting configurations arereducible.For each v ∈ V ( G ) and f ∈ F ( G ) define initial charges µ ( v ) = d ( v ) − ν ( f ) = 2 ℓ ( f ) − −
12. After charges are initially assigned, the onlyelements with negative charge are 4-vertices and 5-vertices. Since chorded 5-cycles are forbidden,there is no 3-fan in G and every 4-face is adjacent to only 4 + -faces. The possible arrangements of3-, 4 + -, or 5 + -faces incident to 4- and 5-vertices are shown in Figure 6.Sequentially apply the following discharging rules. Note that, for a vertex v and a face f , wedefine µ i ( v ) and ν i ( f ) to be the charge on v and f , respectively, after applying rule (R i ).(R1) Let v be a 4-vertex and f be a 4 + -face incident to v . If f is adjacent to a 3-face that is alsoincident to v , then f sends charge 1 to v ; otherwise, f sends charge to v .(R2) Let v be a 5-vertex. If f is a 4 + -face incident to v , then f sends charge to v .A face f is a needy face if ν ( f ) <
0; otherwise, f is non-needy .(R3) If v is a 5-vertex incident to a needy 5-face f , then v sends charge to f .11 (a) 4 + + + + (b) 3 5 + + + (c) 3 3 5 + + (d) 3 5 + + (e) 4 + + + + + (f) 3 5 + + + + (g) 3 5 + + + (h) 3 3 5 + + + (i) 3 3 5 + + v v vv v v v v Figure 6: Possible cyclic arrangements of 3-, 4 + -, and 5 + -faces incident to 4- and 5-verticesA vertex v is a needy vertex if µ ( v ) <
0; otherwise, v is non-needy .(R4) If f is a non-needy 5 + -face incident to a needy 5-vertex v , then f sends charge to v .We show that µ ( v ) ≥ v and ν ( f ) ≥ f . Since the total chargewas preserved during the discharging rules, this contradicts the negative charge sum from the initialcharge values. We begin by considering the charge distribution after applying (R1) and (R2).Let v be a vertex. If v is a 4-vertex, then µ ( v ) = − v receives total charge at least 2 fromits neighboring faces by (R1). Furthermore, v is not affected by any rules after (R1), so µ ( v ) ≥ v is a 6 + -vertex, then µ ( v ) ≥ v is not affected by any other rules, so µ ( v ) ≥
0. If v is a5-vertex, then µ ( v ) = − v receives total charge at least 1 from its neighboring faces by (R2).Therefore, for any vertex v , µ ( v ) ≥ f be a face. If f is a 3-face, then ν ( f ) = 0 and f is not affected by any rule, so ν ( f ) = 0. If f is a 4-face, then ν ( f ) = 2. In (R1) and (R2), the only faces that send charge 1 to a single vertexare adjacent to a 3-face. A 4-face adjacent to a 3-face is a chorded 5-cycle, which is forbidden byassumption, so f sends charge at most to each vertex. Since 4-faces are not affected by rules(R3)–(R4), ν ( f ) ≥
0. If f is a 6 + -face, then f has at least as much initial charge as it has incidentvertices. If v is a 4-vertex incident to f , then f sends charge at most 1 to v by (R1) and does notsend any charge to v by rules (R2)–(R4). If v is a 5-vertex incident to f , then f sends charge to v by (R1), and possibly another charge by (R4), and does not send charge to v by (R1) or (R3).Thus f sends charge at most 1 to each incident vertex, and ν ( f ) ≥ f is a 5-face, then ν ( f ) = 4 and f sends charge at most 1 to each incident vertex by (R1)and (R2). Observe that if ν ( f ) = −
1, then f is incident to five 4-vertices and f is adjacent to atleast one 3-face; this forms (C9), a contradiction. Therefore, we have the following claim about thestructure of a needy 5-vertex. Claim 4.2. If f is a needy 5-face, then ν ( f ) = − and f is adjacent to exactly one -vertex. We now consider the charge distribution after applying (R3). If f is a needy 5-face, then ν ( f ) = − and f is adjacent to exactly one 5-vertex, so ν ( f ) = 0. No faces lose charge in (R3),therefore ν ( f ) ≥ f . 12 laim 4.3. If v is a needy 5-vertex, then v is incident to three 3-faces, two + -faces, and exactlyone needy 5-face; hence µ ( v ) = − .Proof. Suppose that v is a vertex such that µ ( v ) <
0, and consider the cyclic arrangement of 3-and 4 + -faces about v . Case 1: v is incident to at least four + -faces (Figures 6(e) and 6(f )). Since µ ( v ) ≥ µ ( v ) < v is incident to at least three needy 5-faces. Hence two of the needy 5-faces areadjacent, forming (C13), a contradiction. Case 2: v is incident to two non-adjacent -faces and three + -faces (Figure 6(g)). Since µ ( v ) = and µ ( v ) < v is incident to two needy 5-faces, f and f . If these two faces are adjacent,then they form (C13), a contradiction. Otherwise, they share a 3-face t as a neighbor and allvertices incident to f , f , and t other than v are 4-vertices, so the vertices incident to f and t form (C10), a contradiction. Case 3: v is incident to two adjacent -faces and three + -faces (Figure 6(h)). Since µ ( v ) = and µ ( v ) < v is incident to two needy 5-faces, f and f . If f and f are adjacent then theyform (C13), a contradiction. Thus, f and f are not adjacent, but they are each adjacent toa 3-face incident to v . Since f i is needy for each i ∈ { , } , f i sent charge 1 to every 4-vertexincident to f i . By (R1), every 4-vertex incident to f i is incident to a 3-face adjacent to f i .Therefore, f is adjacent to a 3-face that does not share any vertices with the the two 3-facesincident to v , forming one of (C20) or (C21), a contradiction. Case 4: v is incident to three -faces and two + -faces (Figure 6(i)). If v is incident to two needy5-faces f and f , then the 3-face t adjacent to both f and f is incident to two 4-vertices, andthe vertices incident to f and t form (C10), a contradiction. Therefore, v is incident to exactlyone needy 5-face, as claimed.By (R4), every needy 5-vertex receives charge from its unique incident non-needy 5 + -face, so µ ( v ) ≥ v . Each needy 5-face has nonnegative charge after (R3), so if ν ( f ) < f , then f sends charge by (R4), and thus is non-needy. ff t t t v v v v v (a) A 5-face f with ν ( f ) < f v i v i +1 utg i +1 ag i (b) Claim 4.5, Case 1. f v i v i +1 tu wb g i +1 g i (c) Claim 4.5, Case 2. Figure 7: Special cases for a 5-face f with ν ( f ) < f is a 5-face with ν ( f ) < f is incident to vertices v , . . . , v , v is a needy 5-vertex, and f is the needy 5-face incident to v . Let t and t be the adjacentpair of 3-faces incident to v with t adjacent to f and t adjacent to f ; let t be the other 3-faceincident to v . We make two basic claims about this arrangement. Claim 4.4.
The vertex v adjacent to v and incident to t is a + -vertex.Proof. If v is a 4-vertex, then the vertices incident to f and t form (C10), a contradiction. Claim 4.5. If v i and v i +1 are consecutive vertices on the border of f , then at most one of v i and v i +1 is needy.Proof. Suppose that two consecutive vertices v i and v i +1 are needy 5-vertices. Let g i and g i +1 bethe needy 5-faces incident to v i and v i +1 , respectively. Since both v i and v i +1 have three incident3-faces, f is adjacent to a 3-face t across the edge v i v i +1 . Let u be the third vertex incident to t and consider two cases. Case 1: t is not in a diamond (Figure 7(b)). Since g i is needy, the vertex a adjacent to u andincident to g i (with a = v i ) is a 4-vertex and is incident to a 3-face t i such that t i is adjacent to g i . The vertices incident to g i , g i +1 , t , and t i form one of (C15) or (C19), a contradiction. Case 2: t is in a diamond (Figure 7(c)). Let w be the fourth vertex in the diamond and assume,without loss of generality, that v i is adjacent to w . Let b be the vertex incident to g i +1 that isnot adjacent to u or v i +1 along the boundary of g i +1 ; since g i +1 is needy, there is a 3-face t i +1 incident to b and adjacent to g i +1 . The vertices v i and w and those incident to g i +1 and t i +1 form one of (C17) or (C18), a contradiction.By Claim 4.5, f is incident to at most two needy vertices, and by Claim 4.4, v is non-needy. If f is incident to exactly one needy 5-vertex, then v , v , and v are 4-vertices since µ ( f ) = 0, butthen the vertices incident to f and f form (C14), a contradiction.Therefore, f is incident to two needy vertices, and since v is a 5 + -vertex, f is incident to exactlytwo 4-vertices. Each of these receives charge 1, so ν ( f ) = − . By Claim 4.5, the needy verticesincident to f consist of v and exactly one of v or v . The needy 5-vertex v i other than v is alsoincident to three 3-faces t , t , and t , where t and t form a diamond with t adjacent to f . ByClaim 4.4, the vertex adjacent to v i and incident to both f and t is a non-needy 5 + -vertex. Theonly non-needy 5 + -vertex incident to f is v , and hence v is a needy 5-vertex and t is incident to v . If v is a 6 + -vertex, then ν ( f ) ≥
0. Therefore, there is a unique arrangement of needy vertices,4-vertices, and a 5-vertex about a 5-face f with ν ( f ) < i ∈ { , } , let f i be theneedy 5-face incident to the needy 5-vertex v i .The vertices incident to f , f , f , t , and t form (C16), so this arrangement does not appearwithin G ; hence ν ( f ) ≥ f . Therefore, every vertex and face has nonnegative chargeafter (R4), contradicting the negative initial charge sum. Thus, a minimal counterexample doesnot exist and every plane graph with no chorded 5-cycle is (4 , t t t t t f f fv v v v v Figure 8: A non-needy 5-vertex v incident to a non-needy 5-face f with ν ( f ) < In this section we show the case of forbidding chorded 6-cycles from Theorem 1.7. The case offorbidding doubly-chorded 6- and 7-cycles follows from a very similar argument. We give the fullproof for no chorded 6-cycles and describe the differences for the proof when we forbid doubly-chorded 6- and 7-cycles.
Theorem 5.1. If G is a plane graph not containing a chorded 6-cycle, then G is (4 , -choosable.Proof. Let G be a counterexample minimizing n ( G ) among all plane graphs avoiding chorded 6-cycles with a (4 , L such that G is not L -choosable. Observe that n ( G ) ≥
5; infact, δ ( G ) ≥
4. Since G is a minimal counterexample, G does not contain any of the reducibleconfigurations. Specifically, we use the fact that G avoids (C3) and (C4) (see Figure 2).For each v ∈ V ( G ) and f ∈ F ( G ) define initial charge µ ( v ) = d ( v ) − ν ( f ) = ℓ ( f ) − −
8. Since δ ( G ) ≥
4, the only elements of negativecharge are 3-faces. Since a chorded 6-cycle is forbidden and δ ( G ) ≥
4, the clusters (see Figure 1)are triangles (K3), diamonds (K4), 3-fans (K5a), 4-wheels (K5b), and 4-fans with end verticesidentified (K5c). Specifically note that the 4-fan (K6b) contains a chorded 6-cycle, so at most three3-faces in a cluster share a common vertex, unless they form a 4-wheel (K5b) and the commonvertex is the 4-vertex in the center of the wheel.Apply the following discharging rules, as shown in Figure 9.(R1) If f is a 3-face and e is an incident edge, then let g be the face adjacent to f across e .(R1a) If g is a 5 + -face, then f pulls charge from g “through” the edge e .(R1b) If g is a 4-face, then let e , e , and e be the other edges incident to g . For each i ∈ { , , } , let h i be the face adjacent to g across e i . For each i ∈ { , , } , the face f pulls charge from the face h i “through” the edges e and e i .(R2) Let v be a 5 + -vertex, and let f be an incident 3-face.(R2a) If v is a 5-vertex, then v sends charge to f .(R2b) If v is a 6 + -vertex, then v sends charge to f .(R3) If X is a cluster, then every 3-face in X is assigned the average charge of all 3-faces in X .15 f ge f ge h e h e h e v f v f (R1a) (R1b) (R2a) (R2b)Figure 9: Discharging rules in the proof of Theorem 5.1.Notice that the rules preserve the sum of the charges. Let µ i ( v ) and ν i ( f ) denote the chargeon a vertex v or a face f after rule (R i ). We claim that µ ( v ) ≥ v and ν ( f ) ≥ f ; since the total charge sum is preserved by the discharging rules, this contradictsthe negative charge sum from the initial charge values.Let v be a vertex. If v is a 4-vertex, then v is not involved in any rule, so the resulting chargeis 0. If v is a 6 + -vertex, then by (R2b) v loses charge to each incident 3-face. Since G avoidschorded 6-cycles, v is incident to at most ⌊ d ( v ) ⌋ µ ( v ) satisfies µ ( v ) ≥ d ( v ) − − (cid:22) d ( v ) (cid:23) ≥ d ( v ) − − · d ( v ) = 23 d ( v ) − ≥ . If v is a 5-vertex, then by (R2a) v loses charge to each incident 3-face. Since G avoids chorded6-cycles, v is incident to at most three 3-faces, so µ ( v ) ≥ d ( v ) − − · d ( v ) − . Therefore, µ ( v ) ≥ v .Let f be a face. Since 4-faces are not adjacent to 4-faces, (R1b) does not affect the charge valueon 4-faces. Thus, ν ( f ) = 0 for every 4-face f .If f is a 6 + -face, then f loses charge at most through each edge by (R1a) or (R1b), so ν ( f ) ≥ ℓ ( f ) − − ℓ ( f ) = 23 ℓ ( f ) − ≥ . Therefore, ν ( f ) ≥ + -face f .Let f be a 5-face. Since G contains no chorded 6-cycles, f is not adjacent to a 3-face. Therefore, f loses no charge by (R1a), but could lose charge using (R1b), so ν ( f ) ≥ ℓ ( f ) − − ℓ ( f ) = 89 ℓ ( f ) − ≥ . Therefore, ν ( f ) ≥ f is a 5-face. All objects that start with nonnegative charge have nonnegativecharge after the discharging process. It remains to show that each cluster of 3-faces receives enoughcharge to result in a nonnegative charge sum. 16 ase 1: (K3) Let f be an isolated 3-face. The three adjacent faces g , g , and g are all 4 + -faces.By (R1a) or (R1b), f receives charge through each incident edge, so ν ( f ) = − · = 0. Case 2: (K4)
Let f and f be 3-faces in a diamond cluster (K4). Then f is adjacent to two4 + -faces g and g , and f is adjacent to two 4 + -faces h and h . By (R1a) or (R1b), thecluster receives charge through each of the four edges on the boundary of the diamond. Since ν ( f ) + ν ( f ) = −
2, the charge value on the diamond after rule (R1) is − . Since G contains no(C3), there is a 5 + -vertex v incident to both f and f . If v is a 5-vertex, then by (R2a), f and f each receive charge , and the resulting charge on the diamond is zero. If v is a 6 + -vertex,then by (R2b), f and f each receive charge , and the resulting charge on the diamond ispositive. Case 3: (K5a)
Let f , f , and f be 3-faces in a 3-fan cluster (K5a), where f is adjacent to both f and f . The initial charge on this cluster is −
3. There are five edges on the boundary of thiscluster, so by (R1) the cluster receives charge , resulting in charge − after (R1). Note thatthe face f is adjacent to both f and f . Since G contains no (C3), there exists a 5 + -vertex v incident to both f and f , and there exists a 5 + -vertex u incident to both f and f . If v = u ,then by (R2) v sends charge at least to each of f and f and u sends charge at least to eachof f and f , resulting in a nonnegative charge on the 3-fan. If v = u and v is a 6 + -vertex, thenby (R2b) v sends charge to each face f , f , and f , resulting in a nonnegative charge on the3-fan. Otherwise, suppose that v = u and v is a 5-vertex. Since G contains no (C4), there existsanother 5 + -vertex w incident to at least one of f and f . By (R2a) v sends charge to each of f , f , and f , and by (R2) w sends charge at least to at least one of f and f , resulting in anonnegative charge on the 3-fan. Case 4: (K5b)
Let f , f , f , and f be 3-faces in a 4-wheel (K5b). The initial charge on thiscluster is −
4. There are four edges on the boundary of this cluster, so by (R1) the cluster receivescharge , resulting in charge − after (R1). Let v be the 4-vertex incident to all four 3-faces.Let u , u , u , and u be the vertices adjacent to v , ordered cyclically such that vu i u i +1 is theboundary of the 3-face f i for i ∈ { , , } and vu u is the boundary of f . Since G containsno (C3) and d ( v ) = 4, each u i is a 5 + -vertex. By (R2), each u i sends charge at least to thecluster, resulting in a nonnegative total charge. Case 5: (K5c)
Let f , f , f , and f be 3-faces in a 4-strip with identified vertices as in (K5c).The initial charge on this cluster is −
4. Let v , u , u , u , and u be the vertices in the 4-strip,where v is incident to only f and f , u is incident to only f and f , u is incident to f , f ,and f , u is incident to f , f , and f , and u is incident to only f and f . There are six edgeson the boundary of this cluster, so by (R1) the cluster receives charge , resulting in charge − = − f and f form a diamond, and G contains no (C3), one of u and u is a 5 + -vertex.Without loss of generality, assume u is a 5 + -vertex. Since f and f form a diamond, and G contains no (C3), one of u and u is a 5 + -vertex. If u is a 5 + -vertex, then by (R2), thecluster receives charge at least + from u and u , which results in nonnegative total charge.Otherwise, u is a 4-vertex and u is 5 + -vertex. If u is a 6 + -vertex, then by (R2), the clusterreceives charge at least + from u and u . If u is a 5-vertex, then since f and f form adiamond and G contains no (C4), one of v and u is a 5 + -vertex. By (R2), the cluster receives17harge at least + + from u and u and one of v and u . In either case, the final charge isnonnegative.We have verified that the total charge after discharging is nonnegative, contradicting the neg-ative initial charge sum. Thus, a minimal counterexample does not exist and every planar graphwith no chorded 6-cycle is (4 , Corollary 5.2. If G is a plane graph not containing a doubly-chorded 6-cycle or a doubly-chorded7-cycle, then G is (4 , -choosable.Proof. Let G be a minimal counterexample by minimizing n ( G ). Observe that n ( G ) ≥ δ ( G ) ≥
4. Since G contains no doubly-chorded 6-cycle, the clusters are 3-faces (K3), diamonds(K4), 3-fans (K5a), 4-wheels (K5b), and 4-fans with end vertices identified (K5c).Use the same discharging argument as in Theorem 5.1, with the following changes: • If f is a 4-face, then f can be adjacent to a 4-face g . However, since G contains no doubly-chorded 7-cycle, g cannot be adjacent to a 3-face. Therefore, f does not lose charge by rule(R1b). • If f is a 5-face, then f can be adjacent to at most one 3-face g , since G contains no doubly-chorded 7-cycle. By (R1a) f loses charge across the edge it shares with g , and by (R1b) f loses charge at most across the other four edges. Thus ν ( f ) ≥ ℓ ( f ) − − − ≥ . All of the other arguments from the proof of Theorem 5.1 hold, which shows that the resultingtotal charge is nonnegative, and hence a minimal counterexample does not exist.
Theorem 6.1. If G is a plane graph not containing a chorded 7-cycle, then G is (4 , -choosable. We prove the following strengthened statement:
Theorem 6.2.
Let G be a planar graph with no chorded 7-cycle, and let P be a subgraph of G , where P is isomorphic to one of P , P , P , or K , and all vertices in V ( P ) are incident to a commonface f . Let L be a (4 , -list assignment of G − P and let c be a proper coloring of P . There existsan extension of c to a proper coloring of G such that c ( v ) ∈ L ( v ) for all v ∈ V ( G − P ) .Proof. Suppose that there exists a counterexample. Select a counterexample (
G, P, L, c ) by mini-mizing n ( G ) − n ( P ) among all chorded 7-cycle free plane graphs, G , with a subgraph P isomorphicto a graph in { P , P , P , K } , a proper coloring c of P , and a (4 , L of G − P such that c does not extend to an L -coloring of G . We will refer to the vertices of P as precoloredvertices . Claim 6.3. G is 2-connected. roof. If G is disconnected, then each connected component can be colored separately. Supposethat G has a cut-vertex v . Then there exist connected subgraphs G and G where G = G ∪ G and V ( G ) ∩ V ( G ) = { v } , n ( G ) < n ( G ), and n ( G ) < n ( G ). We can assume without loss ofgenerality that G contains at least one vertex of P , so let S be the subgraph of P contained in G .If G contains at least one vertex of P , i.e., if v ∈ S , then let S be the subgraph of P containedin G ; otherwise, let S be the vertex v .Since ( G, P, L, c ) is a minimal counterexample, there is an L -coloring c of G that extends thecoloring on S . Using the color prescribed by c on v , there exists an L -coloring c of G thatextends the coloring on S . The colorings c and c form an L -coloring of G , a contradiction. Claim 6.4. G has no separating 3-cycles.Proof. Suppose that P ′ = v v v is a separating 3-cycle of G . Let G be the subgraph of G givenby the exterior of P ′ along with P ′ , and let G be the subgraph of G given by the interior of P ′ along with P ′ . Since P ′ is separating, n ( G ) < n ( G ) and n ( G ) < n ( G ).Since the vertices in P share a common face, we can assume without loss of generality that V ( P ) ⊆ V ( G ). Since ( G, P, L, c ) is a minimal counterexample, there exists an L -coloring c of G .Assign the colors from c to P ′ . Then there exists an L -coloring of G extending the colors on P ′ ,and together c and c form an L -coloring of G , a contradiction. Claim 6.5. If v ∈ V ( P ) such that V ( P ) ⊆ N [ v ] , then the subgraph of G induced by N ( v ) is notisomorphic to any graph in { P , P , P , K } .Proof. Suppose that there exists a vertex v ∈ V ( P ) where all precolored vertices are in N [ v ] andthe subgraph G [ N ( v )] is isomorphic to a subgraph in { P , P , P , K } . Then consider the graph G ′ = G − v with subgraph P ′ = G [ N ( v )]. Since | N G [ v ] | ≤
4, there exists an L -coloring c ′ of G [ N [ v ]]. Since ( G, P, L, c ) is a minimal counterexample, c ′ extends to an L -coloring of G ′ , whichin turn extends to an L -coloring of G , a contradiction. Claim 6.6. If v ∈ V ( P ) has d G ( v ) ≤ , then d G ( v ) = 2 and P is isomorphic to P , P , or P .Proof. By Claim 6.3, d G ( v ) = 1. If d G ( v ) = 2 and P ∼ = K , then G [ N G ( v )] is isomorphic to P ,contradicting Claim 6.5. Claim 6.7. P is isomorphic to one of P or K .Proof. Suppose that P is not isomorphic to either P or K . If P is isomorphic to P , thenthe vertex of P has two consecutive neighbors u and u not in P ; let U = { u , u } . If P isisomorphic to P , then some vertex v in P has a neighbor u not in P that shares a face withthe edge in P ; let U = { u } . Let P ′ be the subgraph isomorphic to P or K given by includingvertices in U . There exists a proper coloring c ′ of P ′ that extends the coloring on P . But then( G, P ′ , L, c ′ ) has n ( G ) − n ( P ′ ) < n ( G ) − n ( P ), so there exists an L -coloring of G that extends c ′ , a contradiction. Claim 6.8. If v ∈ V ( G − P ) , then d G ( v ) ≥ . roof. Suppose that v ∈ V ( G − P ) has degree d ( v ) ≤
3. Then G − v is a planar graph with nochorded 7-cycle containing a precolored subgraph P and a list assignment L . Since ( G, P, L, c ) is aminimum counterexample, G − v has an L -coloring. However, v has at most three neighbors andat least four colors in the list L ( v ). Thus, there is an extension of the L -coloring of G − v to an L -coloring of G , a contradiction.Observe that n ( G ) ≥
4. Recall that in a configuration (
C, X, ex), an L -coloring of V ( C ) \ X extends to all of C . Because of this fact, if G contains a reducible configuration ( C, X, ex), thenthere is a precolored vertex in the set X , or else G − X has an L -coloring that extends to all of G .Specifically, we will use the fact that G avoids (C2), (C3), (C4), (C5), (C6), (C7), and (C8).For each v ∈ V ( G ) and f ∈ F ( G ) define µ ( v ) = d ( v ) − δ ( v ) and ν ( f ) = ℓ ( f ) − ε ( f ) , where δ ( v ) ∈ { , } has value 1 if and only if v ∈ V ( P ), and ε ( f ) ∈ { , } has value 1 if and onlyif the boundary of f is the set of precolored vertices, V ( P ). By Euler’s Formula, the initial chargesum is at most −
1. Claims 6.6 and 6.8 assert that the only negatively-charged objects are 3-faces.For a vertex v , let t k ( v ) denote the number of k -faces incident to v . Apply the followingdischarging rules. Let µ i ( v ) and ν i ( f ) denote the charge on a vertex v or a face f after rule (R i ). f ge v f v f (R1a) (R2a) (R2b) f ge h e h e h e f ge h e h e f f ge h e h e f (R1b) (R1c), Case 1 (R1c), Case 2Figure 10: Discharging rules (R1) and (R2) in the proof of Theorem 6.1.(R0) If v is a precolored vertex and f is an incident 3-face with negative charge, then v sendscharge to f .(R1) If f is a 3-face and e is an incident edge, then let g be the face adjacent to f across e .20R1a) If g is a 5 + -face, then f pulls charge from g “through” the edge e .(R1b) If g is a 4-face and f is the only 3-face adjacent to g , then let e , e , and e be the otheredges incident to g . For each i ∈ { , , } , let h i be the face adjacent to g across e i . Foreach i ∈ { , , } , the face f pulls charge from the face h i “through” the edges e and e i .(R1c) If g is a 4-face and g is adjacent to two 3-faces f and f (say f = f ), then let e and e be the other edges incident to g , where the faces h and h sharing these edges are6 + -faces. For each i ∈ { , } , the face f pulls charge from the face h i “through” theedges e and e i .(R2) Let v be a 5 + -vertex with v / ∈ V ( P ) and let f be an incident 3-face.(R2a) If v is a 5-vertex, then v sends charge a to f , when a = max { , t ( v ) } .(R2b) If v is a 6 + -vertex, then v sends charge to f .(R3) If f is a 6-face with ν ( f ) < v is an incident 5 + -vertex or an incident vertex in V ( P )with µ ( v ) >
0, then v sends charge to f .We claim that µ ( v ) ≥ v and ν ( f ) ≥ f . Since the totalcharge sum was preserved during the discharging rules, this contradicts the negative charge sumfrom the initial charge values.Note that 6-faces are not incident to 3-faces since G does not contain a chorded 7-cycle. Observethat a 6-face f has ν ( f ) < f are 4-faces, and each of those4-faces has two adjacent 3-faces. Claim 6.9.
Let v be a vertex in V ( P ) . Then µ ( v ) ≥ . In addition, if v is incident to a -face f with ν ( f ) < , then µ ( v ) > .Proof. By Claims 6.6 and 6.7, we have µ ( v ) = d ( v ) − ≥
0. Note that if µ ( v ) ≥ t ( v ) + t ( v ),then the final charge µ ( v ) is nonnegative. Since d ( v ) ≥ t ( v ) + t ( v ), it suffices to show that µ ( v ) ≥ d ( v ) + t ( v ). Case 1: P ∼ = P . Let v , v , and v be the vertices in the 3-path P . For i ∈ { , , } , µ ( v i ) = d ( v i ) −
2. Since P is not isomorphic to K , these vertices do not form a cycle, and the faceto which all vertices are incident is not a 3-face. Hence t ( v i ) ≤ d ( v i ) −
1. If d ( v i ) ≥
4, then µ ( v i ) = d ( v i ) − ≥ d ( v i ) > d ( v i ) + t ( v i ).If d ( v i ) = 2, then µ ( v i ) = 0. If i = 2, then v is not incident to any 3-faces since v and v are not adjacent. If i ∈ { , } and v i is adjacent to a 3-face, then let v ′ i be the neighbor of v i not in V ( P ). Let P ′ be the subgraph induced by ( V ( P ) ∪ { v ′ i } ) \ { v i } , which forms a copy of P or K in G − v i . For any color c ( v ′ i ) ∈ L ( v ′ i ) \ { c ( v i ) } , there exists an L -coloring of G − v i as ( G − v i , P ′ , L, c ) is not a counterexample; this coloring extends to an L -coloring of G . Thus, t ( v i ) = 0. If v i is incident to a 6-face f with ν ( f ) <
0, then the other face incident to v i is a4-face that is adjacent to two 3-faces. This results in a chorded 7-cycle, a contradiction; thus(R3) does not apply to v i .If d ( v i ) = 3, Claim 6.4 asserts that G has no separating 3-cycles, so then v i loses charge at most1 in (R0). If v i is incident to a 6-face f with ν ( f ) <
0, then the other two faces incident to v i v i and µ ( v i ) ≥ Case 2: P ∼ = K . Let v , v , and v be the vertices in the 3-cycle P , so µ ( v i ) = d ( v i ) − v i .By Claim 6.4, G has no separating 3-cycle, so the three vertices are incident to a common 3-face f with ν ( f ) = 0. Therefore, each vertex v i sends charge to at most d ( v i ) − d ( v i ) ≥ d ( v i ) = 3. If t ( v i ) >
1, the subgraphof G induced by the neighborhood of v i is isomorphic to P or K , contradicting Claim 6.5. If d ( v i ) ≥
4, then µ ( v i ) = d ( v i ) − ≥ d ( v i ) ≥ d ( v i ) + t ( v i ). Therefore, µ ( v i ) ≥ v has µ ( v ) ≥ f is a 4-face, then (R1b) and (R1c) do not pull charge from f , since this would require f tobe adjacent to a 4-face g that is adjacent to a 3-face t , but then f , g , and t form a doubly-chorded7-cycle. Thus, ν ( f ) = 0 for every 4-face f .If f is a 5-face, then since G contains no chorded 7-cycles, f is not adjacent to two 3-faces and f is not adjacent to a 4-face. Therefore, f loses charge at most by (R1a), but loses no chargeusing (R1b), so ν ( f ) > f .If f is a 6-face, then f is not adjacent to a 3-face since G contains no chorded 7-cycle. Observethat by Claim 6.3 the boundary of f is a simple 6-cycle. So if f sends charge through an edge e during (R1), it can send charge through e by (R1b), or it can send charge through e by (R1c).The only way that this will result in a negative charge after (R1) and (R2) is for f to send charge through each of its six edges by (R1c); this will cause ν ( f ) = 2 − · = − . If f has a precoloredvertex v on its boundary, then by Claim 6.9, v has positive charge after (R0); by (R3), f receivescharge at least , resulting in ν ( f ) ≥
0. If f has no incident precolored vertices, then since G contains no (C2), some vertex v on the boundary of f is a 5 + -vertex. By (R3) v sends charge to f and hence ν ( f ) ≥
0. Observe the following claim concerning the structure about a vertex thatloses charge by (R3).
Claim 6.10.
Let v be a + -vertex with the three incident faces f , f , and f , in cyclic order. If v sends charge to f by (R3), then f and f are 4-faces and f is a 6-face. If f is a 7 + -face, then by (R1) f loses charge at most through each edge. Thus, ν ( f ) ≥ ℓ ( f ) − − ℓ ( f ) = 58 ℓ ( f ) − > . Therefore, ν ( f ) > + -face f .Next, we will consider a vertex v not in V ( P ).If v is a 4-vertex, then v does not lose charge by any rule, so the resulting charge is 0.If v is a 5-vertex, let a = max { , t ( v ) } and v loses charge a t ( v ) to incident 3-faces by (R2a). If(R3) does not apply to v , then v sends charge at most 1 to incident 3-faces and µ ( v ) ≥
0. If (R3)applies to v , then v is incident to faces f , f , and f where f and f are 4-faces and f is a 6-face.Since d ( v ) = 5 and G has no chorded 7-cycle, the rule (R3) applies at most once. If (R3) appliesonce, then t ( v ) ≤ v loses charge at most by (R2) and charge by (R3), so µ ( v ) ≥ v is a 6 + -vertex, then let k = t ( v ) and ℓ be the number of times (R3) applies to v . Noticethat k ≤ ⌊ d ( v ) ⌋ since G avoids chorded 7-cycles. Further, notice that k + 2 ℓ ≤ d ( v ), since each22-face that gains charge from v by (R3) is preceded by a 4-face in the cyclic order of faces around v . By (R2b), v can lose charge to each incident 3-face, and v can lose charge at most to eachincident 6-face by (R3). Then v ends with charge µ ( v ) ≥ d ( v ) − − k − ℓ. If d ( v ) = 6, then observe k + ℓ ≤ µ ( v ) ≥
0. If d ( v ) = d ≥
7, then d , k , and ℓ satisfythe following linear program with dual on variables a , a , and a :min d − k − ℓ s.t. d ≥ d − k ≥ d − k − ℓ ≥ d, k, ℓ ≥ a s.t. a + 5 a + a ≤ − a − a ≤ − − a ≤ − a , a , a ≥ a , a , a ) = (cid:0) , , (cid:1) demonstrates that d − k − ℓ ≥ · >
4, andthus µ ( v ) > + -vertex.It remains to be shown that the clusters receive enough charge to become nonnegative. Since G contains no separating 3-cycle, G does not contain the cluster (K5c) or the clusters (K6g)–(K6r).Observe that there is no precolored vertex v of degree at most three where all faces incident to v have length three. Finally, it is worth noting again that if G contains a reducible configuration( C, X, ex), then there is a precolored vertex in the set X .If a vertex v is a 5 + -vertex or v ∈ V ( P ), we say v is full ; if v is a 6 + -vertex or v ∈ V ( P ), then v is heavy . Note that a heavy vertex v sends charge to each incident negatively-charged 3-faceby (R0) or (R2b). If P ∼ = K , we call P the precolored face. f vf f f f f (K3) (K4) (K5a)Figure 11: Clusters (K3), (K4), and (K5a) Case 1: (K3)
Let f be the isolated 3-face in (K3). If f is the precolored face, then ν ( f ) = ν ( f ) = 0.Otherwise, the initial charge on f is − . By (R1), f receives charge through its boundary edges,resulting in a nonnegative final charge. Case 2: (K4)
Let f and f be 3-faces in a diamond cluster (K4). First, suppose without loss ofgenerality that f is the precolored face. The initial charge of the cluster is −
1. Then f receivescharge 1 by (R0) and charge 2 · by (R1), resulting in a positive final charge. Otherwise, theinitial charge on the cluster is − . By (R1), f and f receive charge through each of the twoedges on the boundary of the cluster, resulting in charge − . If the cluster contains a precoloredvertex u , then it receives charge by (R0). Otherwise, since G contains no (C3), there is a 5 + -vertex v incident to both f and f . By (R2), this vertex sends charge at least to each of thefaces, resulting in a nonnegative final charge. Case 3: (K5a)
Let f , f , and f be 3-faces in a 3-fan cluster (K5a), where f is adjacent to both f and f . Suppose that the cluster contains a precolored face, so the initial charge on the cluster23s −
2. If f is precolored, then the cluster receives charge 4 · by (R0); if f or f is precolored,then the cluster receives charge 3 · by (R0) and charge 3 · by (R1). In either case, the finalcharge is nonnegative.If P = K or the cluster does not contain the precolored face, then the initial charge on thecluster is − . By (R1), the cluster receives charge 5 · , resulting in charge − . Note that thefaces f and f form a diamond and the faces f and f form a diamond. Since G contains no(C3), there exists a full vertex v incident to both f and f . Similarly, there exists a full vertex u incident to f and f . If u = v, then by (R0) or (R2), v sends charge at least to each of f and f and u sends charge at least to each of f and f , resulting in nonnegative charge onthe cluster. If u = v and v is a heavy vertex, then v sends charge to each face f , f , and f ,resulting in nonnegative charge on the cluster. Otherwise, suppose that u = v / ∈ V ( P ) and v isa 5-vertex. Since G contains no (C4), there exists another full vertex w that is incident to atleast one of f and f . By (R2a), v sends charge to f , f , and f , and by (R0) or (R2), w sends charge at least to one of f and f , resulting in nonnegative charge on the cluster. v u u u u f f f f f f f f v u u u u u f f f f (K5b) (K6a) (K6b)Figure 12: Clusters (K5b), (K6a), and (K6b) Case 4: (K5b)
Let f , f , f , and f be 3-faces in a 4-wheel (K5b). If the cluster contains aprecolored face, then the initial charge on the cluster is −
3; the cluster receives charge 5 · by(R0) and charge 3 · by (R1), resulting in a positive final charge. Otherwise, the initial chargeon this cluster is −
4. By (R1), the cluster receives charge 4 · , resulting in charge − . Let v be the 4-vertex incident to all four 3-faces. Let u , u , u , and u be the vertices adjacent to v ,ordered cyclically such that vu i u i +1 is the boundary of the 3-face f i for i ∈ { , , } and vu u isthe boundary of f . Since the cluster does not contain the precolored face, v is not a precoloredvertex. Since G contains no (C3), each u i is a full vertex. When u i is a 5-vertex, it is incident totwo 7 + -faces, so u i sends charge to each incident 3-face by (R2). Thus, each u i sends chargeat least 2 · to the cluster by (R0) or (R2), resulting in a nonnegative final charge. Case 5: (K6a)
Let f , f , f , and f be 3-faces in a 4-strip cluster (K6a). If the cluster containsthe precolored face, then the initial charge on the cluster is −
3. If f or f is precolored, thenthe cluster receives charge 3 · by (R0) and charge 4 · by (R1); if f or f is precolored, thenthe cluster receives charge 5 · by (R0) and charge 5 · by (R1). In either case, the resultingfinal charge is nonnegative. If the cluster does not contain the precolored face, then the initialcharge on this cluster is −
4. By (R1), the cluster receives charge 6 · , resulting in charge − .Note that for i ∈ { , , } , the faces f i and f i +1 form a diamond. Since G contains no (C3),there exists a full vertex v incident to both f i and f i +1 . Let u be a full vertex incident to f and f . Without loss of generality, u is not incident to f , so there is a full vertex u incidentto f and f . If u is a heavy vertex, the cluster receives charge 3 · from u by (R0) or (R2b),and charge at least 2 · from u by (R0) or (R2), resulting in a positive final charge. Otherwise,24 is a 5-vertex, so u sends charge 3 · by (R2a), resulting in charge − . If u is incidentto f , then u sends charge at least 3 · by (R0) or (R2), resulting in a positive final charge.Otherwise, u is incident with f and f but not f . If u is a large vertex, it sends charge 2 · by (R0) or (R2b). Otherwise, since G contains neither a (C3) or a (C4), there is a third fullvertex u . The cluster receives charge 2 · from u by (R2a) and charge at least from u by(R0) or (R2). In each case, the resulting final charge is nonnegative. Case 6: (K6b)
Let f , f , f , and f be 3-faces in a 4-fan cluster (K6b). Let v be the center of thefan, with neighbors u , u , u , u , and u where for i ∈ { , , } , f i and f i +1 are adjacent on theedge vu i +1 . If the cluster contains the precolored face, then the initial charge on the cluster is −
3. If f or f is precolored, then the cluster receives charge 4 · by (R0) and charge 4 · by(R1); if f or f is precolored, then the cluster receives charge 5 · by (R0) and charge 5 · by(R1). In either case, the resulting final charge is positive.If the cluster does not contain the precolored face, then the initial charge on this cluster is − · , resulting in charge − . If v is a heavy vertex, thenby (R0) or (R2b) v sends charge 4 · to the cluster, resulting in positive charge. Otherwise, v / ∈ V ( P ) and v is a 5-vertex, so v sends charge 1 to the cluster by (R2a), resulting in charge − . If there is a heavy vertex in { u , u , u } , then that vertex contributes charge 2 · to thecluster, resulting in a positive charge. If there is no heavy vertex in { u , u , u } , then there isat least one 5-vertex in { u , u , u } since G contains no (C4). If there are multiple 5-vertices in { u , u , u } , then each sends charge 2 · to the cluster by (R2a), resulting in positive charge. Ifthere is only 5-vertex w among u , u , and u , then there is a full vertex z ∈ { u , u } since G does not contain (C4) or (C5); the cluster receives charge 2 · from w by (R2a) and at least from z by (R0) or (R2), resulting in positive final charge. u u u u u u f f f f v u u u u wf f f f g (K6c) (K6d)Figure 13: Clusters (K6c) and (K6d). Case 7: (K6c)
Let f , f , f , and f be the 3-faces of this cluster (K6c) where f is adjacent toeach f i for i ∈ { , , } . If the cluster contains the precolored face, then the initial charge on thecluster is −
3. If one of f , f or f is precolored, the cluster receives charge 4 · by (R0) andcharge 4 · by (R1). If f is precolored, then the cluster receives charge 6 · by (R0). In eithercase, the resulting final charge is nonnegative.If the cluster does not contain the precolored face, then the initial charge on the cluster is − · , resulting in charge − . Let u , u , u , u , u , and u be the vertices on the boundary of the cluster ordered such that u , u , u are the verticesincident to f and f , f and f , and f and f , respectively. Since G contains no (C3), thereare at least two full vertices in { u , u , u } . By (R0) or (R2), these vertices each send charge atleast 1 to the cluster, resulting in a positive total charge.25 ase 8: (K6d) Let f , f , f , and f be cyclically-ordered 3-faces in a 4-wheel with center vertex v where f i and f i +1 share a common edge for i ∈ { , , , } , where indices are taken modulo 4; let g be a 3-face adjacent to f but not incident to v , completing our cluster (K6d). If the clustercontains the precolored face, then the initial charge on the cluster is −
4. If f or f is precolored,then the cluster receives charge 6 · by (R0) and charge 4 · by (R1). If f is precolored, thenthe cluster receives charge 5 · by (R0) and charge 4 · by (R1). If f is precolored, then thecluster receives charge 7 · by (R0) and charge 5 · by (R1). In each of the above cases, thefinal charge is nonnegative. If g is precolored, then the cluster receives charge 4 · by (R0) andcharge 3 · by (R1), resulting in charge − . Let N ( v ) = { u , u , u , u } where u i is incidentto f i and f i +1 for all i ∈ { , , , } . Since G does not contain (C3), u and u are full vertices.Each of u and u sends charge at least 2 · to the cluster by (R2), resulting in nonnegativecharge.If the cluster does not contain the precolored face, then the initial charge on this cluster is − v / ∈ V ( P ). By (R1), the cluster receives charge 5 · , resulting in charge − . Since G does not contain (C3), u , u , u , and u are full vertices. By (R0) or (R2), the cluster receivescharge at least 2 · from each of u and u and charge at least 3 · from each of u and u ,resulting in a positive final charge. v u u u u u f f f f f u zw u f f (K6e) (K6f)Figure 14: Clusters (K6e) and (K6f). Case 9: (K6e)
Let f , f , f , f , and f be the cyclically-ordered 3-faces in a 5-wheel with centervertex v where f i and f i +1 share a common edge for i ∈ { , , , , } , where indices are takenmodulo 5. Let N ( v ) = { u , u , u , u , u } where u i is incident to f i and f i +1 for i ∈ { , , , , } .If the cluster contains the precolored face, then the initial charge on the cluster is −
4. Thecluster receives charge 6 · by (R0) and charge 4 · by (R1), resulting in a positive final charge.If the cluster does not contain the precolored face, then the initial charge is − v / ∈ V ( P ).By (R1), the cluster receives charge 5 · , and by (R2), the cluster receives charge 1 from v ,resulting in charge − . Since G does not contain (C4) or (C6), there are at least three fullvertices in N ( v ). If N ( v ) contains at least three heavy vertices, then the cluster receives chargeat least 6 · by (R0) or (R2b), resulting in a positive final charge. If N ( v ) contains exactly twoheavy vertices, then the cluster receives charge 4 · by (R0) or (R2b) and charge 2 · from a fullvertex by (R2a), resulting in positive charge. If N ( v ) contains exactly one heavy vertex, thenthe cluster receives charge 2 · by (R0) or (R2b) and charge 2 · from each of two full verticesby (R2a), resulting in positive final charge.If N ( v ) contains no heavy vertices, then there are at least three full vertices in N ( v ). Since G does not contain (C4), there are at least two nonadjacent 5-vertices in N ( v ). Further, since G does not contain (C6), (C7), or (C8), there are at least four 5-vertices in N ( v ). The cluster26eceives charge 2 · from each of these vertices by (R2a), resulting in a positive final charge. Case 10: (K6f )
Let f and f be the interior 3-faces in the two overlapping 4-wheels that makeup the cluster (K6f). Let u and u be the shared vertices of f and f and let z and w be thevertices incident with f and f , respectively, that have not yet been labeled. Since G containsno (C3), at least one of u and u is in V ( P ). Then since all the precolored vertices lie on acommon face, the cluster contains the precolored face, so the initial charge is −
5. If f or f isprecolored, then the cluster receives charge 8 · by (R0) and charge 4 · by (R1), resulting ina positive final charge. If one of the other 3-faces is precolored, then the cluster receives charge6 · by (R0) and charge 3 · by (R1), resulting in charge − . Since G contains no (C3), one of w and z is a non-precolored 5 + -vertex. This vertex sends charge at least 3 · to the cluster by(R2), resulting in a positive final charge.We have verified that the total charge after discharging is nonnegative, contradicting the neg-ative initial charge sum. Thus, a minimal counterexample does not exist and every planar graphwith no chorded 7-cycle is (4 , We proved that, for each k ∈ { , , } , planar graphs with no chorded k -cycles are (4 , G is a planar graph with no chorded 4-cycleand no doubly-chorded 7-cycle, then G is (4,2)-choosable using methods similar to those in thispaper. We were unable to extend these results to prove Conjecture 1.3, that all planar graphs are(4 , Acknowledgments
We thank Ryan R. Martin, Alex Nowak, Alex Schulte, and Shanise Walker for participation in theearly stages of the project.
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A Large Reducible Configurations
In the proof of Theorem 4.1, we demonstrated that no minimal counterexample exists by showing thatthere exists a reducible configuration (
C, X, ex) where G contains a copy of C [ X ] as an induced subgraph(and also the copy agrees with the external degrees). In this appendix, we provide the details that clarifythis assumption. By Lemma 3.2, we can relax the condition that C [ X ] is an induced subgraph. We willdemonstrate that the configurations that appear after some vertices in X are merged (while also preservingthe face lengths, vertex degrees, and lack of chorded 5-cycle) result in reducible configurations.Let ( C, X, ex) be a reducible configuration and let { x , x ′ } , . . . , { x t , x ′ t } be a list of vertex pairs in X .For these configurations, we may identify some 3-cycles and 5-cycles that are required to be 5-faces (in thecontext of the proof of Theorem 4.1). The resulting configuration ( C ′ , X ′ , ex) where C ′ and X ′ are modifiedfrom C and X by merging x i with x ′ i and removing any multiedges or loops that result. We say a list { x , x ′ } , . . . , { x t , x ′ t } is valid for ( C, X, ex) if the resulting configuration ( C ′ , X ′ , ex) may appear in a planargraph of minimum degree at least four containing no chorded 5-cycle. There are three situations that canoccur when we perform this action. Pairs too close:
If some pair { x i , x ′ i } have d ( x i , x ′ i ) ≤
2, then either we create a loop or a multiedge whenmerging x i and x ′ i . This will reduce the degree of the resulting vertex, in addition to possibly shorteningknown 3- and 5-cycles. Since distances only decrease as vertices are merged, a pair failing this property willnot appear in any valid list of pairs. Pairs creating chord:
If merging x i and x ′ i creates a chorded 5-cycle, then this configuration would notappear in the minimal counterexample from Theorem 4.1. Since distances only decrease as vertices aremerged, a pair failing this property will not appear in any valid list of pairs. Reducible pairs:
If merging x i and x ′ i does not fit in the above two cases, then we will demonstrate thatthe resulting configuration is reducible. Even if merging one pair of vertices creates a reducible configuration,we need to check all possible lists of pairs that contain that pair.After considering all pairs that could be identified, observe that in each case there is no set of three ormore vertices where every pair can be identified.In the following tables, we list one of the configurations (C10)–(C21), label the vertices, and list all pairsof vertices into the three categories above. In the case of reducible pairs, we present the contracted graph.Most of these contracted graphs contain a copy of (C1), (C2), (C10), (C11), or (C12). The only exceptions re the contracted graphs derived from (C16), but each of these configurations has an Alon-Tarsi orientationand hence is reducible. (C10) abc d e f Pairs too close: ab , ac , ad , ae , af , bc , bd , be , cd , ce , cf , de , df , ef . Pairs creating chord: bf Reducible pairs:
None remain. (C11) abc d efg
Pairs too close: ab , ac , ad , ae , af , ag , bc , bd , bf , bg , cd , ce , cg , de , df , dg , ef , eg , f g . Pairs creating chord: be , cf Reducible pairs:
None remain. (C12) ab c def
Pairs too close: ab , ac , ad , ae , af , bc , bd , be , bf , cd , ce , cf , de , df , ef . Pairs creating chord:
None remain.
Reducible pairs:
None remain. (C13) g hab c d ef
Pairs too close: ab , ac , ad , ag , ah , bc , bg , bh , cf , cg , ch , de , df , dg , dh , ef , eg , eh . Pairs creating chord: ae , af , bf , bd , cd . ce . Reducible pairs: be (contains (C1)) g ha bec df Contains (C1) on 4-cycle be, f, g, c . C14) abc d i e fgh
Pairs too close: ab , ac , ad , ae , ag , ah , ai , bc , bd , bh , bi , cd , ci , de , dh , di , ef , eg , eh , ei , f g , f g , f i , gh , gi , hi . Pairs creating chord: af , be , ce , ch , df , dg . Reducible pairs: bg (contains (C11)), cf (contains (C11)), bg and cf (contains (C12)). a bgcd i efh ab cfd i e gh a bg cfd i eh Contains (C11) Contains (C11) Contains (C12)after deleting vertex h . after deleting vertex d . after deleting vertex h .(C15) abc d i e fgh Pairs too close: ab , ac , ad , ae , ag , ah , ai , bc , bd , bh , bi , cd , ce , ci , de , df , dh , di , ef , eg , eh , ei , f g , f h , f i , gh , gi , hi . Pairs creating chord: af , ag , be , bf ( bf, a, i, h, g, bf ), bh , cg ( cg, d, i, e, f, cg ), ch , dg . Reducible pairs: bg (contains (C2)), cf (contains (C1)), bg and cf (contains (C1)). a bgcd i e fh ab cfd i e gh a bg cfd i eh Contains (C2) Contains (C1) Contains (C1)on 6-cycle bf, f, e, i, d, c . on 4-cycle cf, e, i, d . on 4-cycle cf, e, i, d . C16) abc dej fkℓ m ghi
Pairs too close: ab , ac , ad , ae , af , ag , ai , aj , ak , am , bc , bd , be , bf , bj , bk , bℓ , bm , cd , ce , cj , cm , de , df , di , ef , eg , eh , ei , ej , f g , f h , f i , f j , gh , gi , hi , jk , jℓ , jm , kℓ , km , ℓm . Pairs creating chord: ah , aℓ , bh , bg , bi , cf , ck , cg , ci , cℓ , dg , dh , dj , dk , dm , ek , eℓ , em , f k , f ℓ , f m , gj , gk , gm , hj , ij , ik , im . Reducible pairs: ch (has Alon-Tarsi orientation), dℓ (symmetricto ch ), hk (has Alon-Tarsi orientation), hm (has Alon-Tarsi orien-tation), hℓ (has Alon-Tarsi orientation), gℓ (symmetric to hk ), iℓ (symmetric to hm ). ab chd ej fkℓ m gi abc dej fℓm ghk iabc dej fk m ghℓ i abc dej fkℓ ghm i (C17) abc dh ef g Pairs too close: ab , ac , ad , ae , af , ag , ah , bc , bd , be , bh , cd , ce , cf , ch , de , df , dg , ef , eg , f g . Pairs creating chord: bf , bg , cg , dh , f h , gh . Reducible pairs:
None remaining. C18) abc dh ef g
Pairs too close: ab , ac , ad , ae , af , ag , ah , bc , bd , be , bh , cd , ce , cf , ch , de , df , dg , dh , ef , eg . Pairs creating chord: bf , bg , cg , eg . Reducible pairs:
None remaining. (C19) abc dj i e fgh
Pairs too close: ab , ac , ad , ae , ag , ah , ai , bc , bc , bh , bi , bj , cd , ci , cj , de , dh , di , dj , ef , eg , eh , ei , f g , f h , f i , gh , gi , hi . Pairs creating chord: af , aj , be , ce , cf ( cf, e, i, d, j ), cg ( cg, h, i, d, j ), ch , df , dg . Reducible pairs: bf (contains (C10)), bg (contains (C2)). a bfcd i egh a bgcdj i efh Contains (C10) Contains (C2)on 5-cycle h, g, bf, e, i andvertex a . on 6-cycle bg, f, e, i, d, c . C20) abc dk ij e fgh
Pairs too close: ab , ac , ad , ae , ah , ai , aj , ak , bc , bd , bi , bk , cd , ci , cj , ck , de , dh , di , dj , ef , eg , eh , ei , ej , f g , f h , f i , f j , gh , gi , hi , hj , ij , ik . Pairs creating chord: af ( af, e, j, d, i ), ag ( ag, f, e, j, i ), be , bf ( bf, a, i, j, e ), bg ( bg, h, i, a, k ), bh , bj , ce , cf ( cf, d, i, j, e ), cg ( cg, h, i, j, d ), ch , df , dg , dk , ek ( ek, j, d, i, a ), f k ( f k, e, j, i, a ), gj , gk ( gk, h, i, a, b ), jk ( jk, d, c, b, a ). Reducible pairs: hk (contains (C10)). abc d hki e fg Contains (C10) on 5-cycle hk, g, f, e, i and vertex a .(C21) abc dk ij e fgh Pairs too close: ab , ac , ad , ae , ah , ai , aj , ak , bc , bd , bi , bk , cd , ci , cj , ck , de , dh , di , dj , dk , ef , eg , eh , ei , ej , f g , f h , f i , f j , gh , gi , hi , hj , ij . Pairs creating chord: af ( af, e, j, d, i ), ag ( ag, f, e, j, i ), be , bf ( bf, a, i, j, e ), bh , bj , ce , cf ( cf, d, i, j, e ), cg ( cg, h, i, j, d ), ch , df , dg , dk , ek ( ek, j, i, d, c ), gj , hk ( hk, i, a, b, c ), ik , jk ( jk, i, a, b, c ). Reducible pairs: f k (Contains (C11)), gk (Contains (C11)), bg and f k (Contains (C12)). (Note: if we identify only bg , then k must be identified with f in order to preserve that g has totaldegree four.) a bcd f ki egh abcd gki efh a bgcd i ef kh Contains (C11) Contains (C11) Contains (C12)after deleting vertices c and d . after deleting vertices c and d . after deleting vertices c and d ..