A degree sequence version of the Kühn-Osthus tiling theorem
aa r X i v : . [ m a t h . C O ] S e p A DEGREE SEQUENCE VERSION OF THE K ¨UHN–OSTHUS TILINGTHEOREM
JOSEPH HYDE AND ANDREW TREGLOWN
Abstract.
A fundamental result of K¨uhn and Osthus [The minimum degree threshold for perfectgraph packings, Combinatorica, 2009] determines up to an additive constant the minimum degreethreshold that forces a graph to contain a perfect H -tiling. We prove a degree sequence version ofthis result which allows for a significant number of vertices to have lower degree. MSC2000: 05C35, 05C70.1.
Introduction
Minimum degree conditions forcing tilings.
A substantial branch of extremal graphtheory concerns the study of tilings . Given two graphs H and G , an H -tiling in G is a collectionof vertex-disjoint copies of H in G . An H -tiling is called perfect if it covers all the vertices of G .Perfect H -tilings are also often referred to as H -factors , perfect H -packings or perfect H -matchings .In the case when H has a component on at least 3 vertices, the decision problem of whethera graph contains a perfect H -tiling is NP-complete [6]. Thus, there has been a focus on estab-lishing sufficient conditions to force a perfect H -tiling. The seminal Hajnal–Szemer´edi theorem [5]characterises the minimum degree that ensures a graph contains a perfect K r -tiling. Theorem 1.1 (Hajnal and Szemer´edi [5]) . Every graph G whose order n is divisible by r andwhose minimum degree satisfies δ ( G ) ≥ (1 − /r ) n contains a perfect K r -tiling. Moreover, thereare n -vertex graphs G with δ ( G ) = (1 − /r ) n − that do not contain a perfect K r -tiling. The following result of Alon and Yuster [1] shows that any sufficiently large graph G withminimum degree slightly above that in Theorem 1.1 in fact contains a perfect H -tiling for anygraph H with χ ( H ) = r . Theorem 1.2 (Alon and Yuster [1]) . Suppose that γ > and H is a graph with χ ( H ) = r . Thenthere exists an integer n = n ( γ, H ) such that the following holds. If G is a graph whose order n ≥ n is divisible by | H | and δ ( G ) ≥ (1 − /r + γ ) n then G contains a perfect H -tiling. For many graphs H the minimum degree condition in Theorem 1.2 is best-possible up to theterm γn . Indeed, for many graphs H there are so-called divisibility barrier constructions G on n vertices that have minimum degree (1 − /χ ( H )) n − H -tiling (see [15,Section 2]). However, Koml´os, S´ark¨ozy and Szemer´edi [12] proved that the term γn in Theorem 1.2can be replaced with a constant dependent only on H . Further, as discussed shortly, K¨uhn andOsthus [14, 15] proved that there are also many graphs H for which one can significantly reducethe minimum degree condition in Theorem 1.2.In a related direction, Koml´os [10] showed that if one only requires an H -tiling covering almost all vertices in the host graph, then one can replace the χ ( H )-term in the minimum degree condition JH: University of Birmingham, United Kingdom, [email protected] .AT: University of Birmingham, United Kingdom, [email protected] . f the Alon–Yuster theorem by the so-called critical chromatic number χ cr ( H ) of H . Here χ cr ( H )is defined as χ cr ( H ) := ( χ ( H ) − | H || H | − σ ( H ) , where σ ( H ) denotes the size of the smallest possible colour class in any χ ( H )-colouring of H . Notethat all graphs H satisfy χ ( H ) − < χ cr ( H ) ≤ χ ( H ) and χ cr ( H ) = χ ( H ) precisely when every χ ( H )-colouring c of H is balanced (i.e. the colour classes of c have the same size). Theorem 1.3 (Koml´os [10]) . Let η > and let H be a graph. Then there exists an integer n = n ( η, H ) such that every graph G on n ≥ n vertices with δ ( G ) ≥ (cid:18) − χ cr ( H ) (cid:19) n contains an H -tiling covering all but at most ηn vertices. Note that the minimum degree condition in Theorem 1.3 is best-possible in the sense that onecannot replace the (1 − /χ cr ( H )) term here with any smaller fixed constant (this is a consequence of[10, Theorem 7]). Further, Koml´os [10] also determined the minimum degree threshold for ensuringa graph G contains an H -tiling covering an x th proportion of its vertices, for any x ∈ (0 , H .K¨uhn and Osthus [14, 15] showed that for many graphs H , a minimum degree slightly above thatin Koml´os’ theorem actually ensures a perfect H -tiling. To state their result we need to introducesome notation. We say that a colouring of H is optimal if it uses exactly χ ( H ) =: r colours.Let C H denote the set of all optimal colourings of H . Given an optimal colouring c of H , let x c, ≤ x c, ≤ · · · ≤ x c,r denote the sizes of the colour classes of c . We write D ( c ) := { x c,i +1 − x c,i | i = 1 , . . . , r − } , and let D ( H ) := [ c ∈ C H D ( c ) . We denote by hcf χ ( H ) the highest common factor of all integers in D ( H ). If D ( H ) = { } thenwe define hcf χ ( H ) := ∞ . We write hcf c ( H ) for the highest common factor of all the orders ofcomponents of H . For non-bipartite graphs H we say that hcf( H ) = 1 if hcf χ ( H ) = 1. If χ ( H ) = 2then we say hcf( H ) = 1 if hcf c ( H ) = 1 and hcf χ ( H ) ≤
2. (See [15] for some examples.) Set χ ∗ ( H ) := ( χ cr ( H ) if hcf( H ) = 1; χ ( H ) otherwise.Also let δ ( H, n ) denote the smallest integer k such that every graph G whose order n is divisibleby | H | and with δ ( G ) ≥ k contains a perfect H -tiling.When hcf( H ) = 1, K¨uhn and Osthus showed that χ cr ( H ) is the parameter governing the min-imum degree condition that ensures a perfect H -tiling. When hcf( H ) = 1, χ ( H ) instead is therelevant parameter. Theorem 1.4 (K¨uhn and Osthus [15]) . For every graph H there exists a constant C = C ( H ) suchthat (cid:18) − χ ∗ ( H ) (cid:19) n − ≤ δ ( H, n ) ≤ (cid:18) − χ ∗ ( H ) (cid:19) n + C. ntuitively speaking, graphs H with hcf( H ) = 1 avoid certain divisibility barrier problems whenseeking a perfect H -tiling, thus ensuring the lower threshold in this case in Theorem 1.4. EarlierK¨uhn and Osthus [14] had proven a version of Theorem 1.4 for graphs H with χ ( H ) ≥ H ) = 1; there though the constant C ( H ) was replaced with a linear error term. We now statethis result explicitly for future reference. Theorem 1.5 (K¨uhn and Osthus [14]) . Let η > and H be a graph with hcf χ ( H ) = 1 and χ ( H ) =: r ≥ . Then there exists an integer n = n ( η, H ) such that the following holds. Let G bea graph on n ≥ n vertices such that | H | divides n and δ ( G ) ≥ (cid:18) − χ cr ( H ) + η (cid:19) n. Then G contains a perfect H -tiling. Degree sequence conditions forcing tilings.
As discussed in the previous subsection, theminimum degree conditions in each of the Hajnal–Szemer´edi theorem, Koml´os’ theorem and theK¨uhn–Osthus theorem are essentially best-possible. However, this does not mean one cannot seeksignificant strengthenings of these results. For example, Kierstead and Kostochka [9] proved an
Ore-type generalisation of Theorem 1.1 where now one replaces the minimum degree condition withthe condition that the sum of the degrees of every pair of non-adjacent vertices in G is at least2(1 − /r ) n − H -tiling. Thestudy of degree sequence results for tilings was initiated in [2]. In particular, a conjecture on adegree sequence strengthening of the Hajnal–Szemer´edi theorem was raised [2, Conjecture 7], aswell as a degree sequence version of the Alon–Yuster theorem [2, Conjecture 8]. In [20] the secondauthor proved the latter conjecture (also yielding an asymptotic version of Conjecture 7 from [2]). Theorem 1.6 (Treglown [20]) . Suppose that η > and H is a graph with χ ( H ) =: r ≥ . Thenthere exists an integer n = n ( η, H ) such that the following holds. If G is a graph whose order n ≥ n is divisible by | H | , and whose degree sequence d ≤ · · · ≤ d n satisfies d i ≥ ( r − n/r + i + ηn for all i < n/r, then G contains a perfect H -tiling. Theorem 1.6 is a significant strengthening of the Alon–Yuster theorem as it allows for n/r verticesto have degree (significantly) below that required in the latter. Further Theorem 1.6 provides thefirst piece of a degree sequence analogue of the K¨uhn–Osthus theorem.The main result in this paper deals with the remaining part of this problem, providing a degreesequence condition that forces a perfect H -tiling for graphs with hcf( H ) = 1. Theorem 1.7.
Let η > and H be a graph with hcf( H ) = 1 and χ ( H ) =: r ≥ . Let σ := σ ( H ) , h := | H | and ω := ( h − σ ) / ( r − . Then there exists an integer n = n ( η, H ) such that thefollowing holds. Let G be a graph on n ≥ n vertices such that h divides n and G has degreesequence d ≤ . . . ≤ d n such that d i ≥ (cid:18) − ω + σh (cid:19) n + σω i + ηn for all ≤ i ≤ ωnh .Then G contains a perfect H -tiling. Observe that when i = ωn/h , we have d ωnh ≥ (cid:16) − ωh + η (cid:17) n = (cid:18) − χ cr ( H ) + η (cid:19) n. ence, Theorem 1.7 is a strengthening of Theorem 1.5. Note that Theorem 1.7 applies to all graphs H with hcf( H ) = 1, not just graphs H with χ ( H ) ≥ χ ( H ) = 1 (as in Theorem 1.5).Moreover, Theorem 1.7 (and Theorem 1.6) is best-possible for many graphs H in the sense thatwe cannot replace the ηn -term with a o ( √ n )-term (see Proposition 2.3). Theorem 1.7 is also bestpossible for all graphs H in the sense that there are n -vertex graphs G with only slightly more than ωn/h vertices with degree (slightly) below (1 − ω/h + η ) n that do not contain a perfect H -tiling(see Proposition 2.6). Thus, it is not possible to allow significantly more ‘small’ degree vertices inTheorem 1.7. Extremal examples are discussed in more detail in Section 2.2.Combining Theorem 1.7 with Theorem 1.6 we obtain the following degree sequence version ofthe K¨uhn–Osthus theorem (Theorem 1.4). Theorem 1.8.
Let η > and H be a graph with χ ( H ) =: r ≥ . Let σ := σ ( H ) , h := | H | and ω := ( h − σ ) / ( r − . Then there exists an integer n = n ( η, H ) such that if G is a graph on n ≥ n vertices, h divides n and either (i) or (ii) below holds, then G contains a perfect H -tiling. (i) hcf( H ) = 1 and G has degree sequence d ≤ . . . ≤ d n such that d i ≥ (cid:18) − ω + σh (cid:19) n + σω i + ηn for all ≤ i ≤ ωnh . (ii) hcf( H ) = 1 and G has degree sequence d ≤ . . . ≤ d n such that d i ≥ ( r − n/r + i + ηn for all i < n/r. One can in fact obtain the following generalisation of Theorem 1.7.
Theorem 1.9.
Let η > and H be a graph with hcf( H ) = 1 and χ ( H ) =: r ≥ . Let h := | H | .Set σ ∈ R such that σ ( H ) ≤ σ < h/r and ω := ( h − σ ) / ( r − . Then there exists an integer n = n ( η, H ) such that the following holds. Let G be a graph on n ≥ n vertices such that h divides n and G has degree sequence d ≤ . . . ≤ d n such that d i ≥ (cid:18) − ω + σh (cid:19) n + σω i + ηn for all ≤ i ≤ ωnh .Then G contains a perfect H -tiling. In Section 7, we will prove Theorem 1.9 directly. The proof of Theorem 1.9 follows that ofTheorem 1.5 in [14] closely. The main novelty of our proof is how we avoid divisibility barriers.For this we make use of an elementary number theoretic result for graphs with hcf( H ) = 1 (seeTheorem 5.2). We also make use of a recent degree sequence strengthening of Koml´os’ theoremproved by the authors and Liu [7].Since the choice of σ ∈ [ σ ( H ) , h/r ) is arbitrary, note that Theorem 1.9 provides an infinitecollection of degree sequences that force a perfect H -tiling. Having a higher value of σ lowers thestarting point of the degree sequence condition, but at the price of a steeper ‘slope’ and highervalue of d ωn/h (see Figure 1). As with Theorem 1.7, for many graphs H , each of these degreesequences is best-possible in the sense that we cannot replace the ηn -term with a o ( √ n )-term (seeSection 2.2). Note too that we cannot extend Theorem 1.9 to the case when σ < σ ( H ). Indeed,in this case, if we set η ≪ G to have degree below (1 − /χ cr ( H )) n −
1; however, we know from Theorem 1.4 thatthere are graphs G that satisfy this condition and that do not contain perfect H -tilings.The paper is organised as follows. In the next section we discuss various senses of optimality fordegree sequence conditions before giving several extremal examples for Theorems 1.7 and 1.9. Wealso ask whether one can improve Theorem 1.7 by suitably ‘capping’ the bounds on the degreesof the vertices (see Question 2.7). In Section 3 we introduce some notation and definitions; in d Degree (cid:0) − r (cid:1) n + ηn (cid:16) − r − σ ( H )+ ω ( H )2 h (cid:17) n + ηn (cid:16) − ω ( H )+ σ ( H ) h (cid:17) n + ηn (cid:16) − ω ( H ) h (cid:17) n + ηn (cid:16) − r − ω ( H )2 h (cid:17) n + ηn (cid:0) − r (cid:1) n + ηn d n r d (cid:16) h + r ω ( H ) r h (cid:17) n d ω ( H ) n h Figure 1.
The degree sequence in Theorem 1.9 for a fixed graph H given σ = σ ( H )(medium dashed); σ = h + rσ ( H )2 r (long dashed); σ = hr (full).Section 4 we give a number of auxiliary results and definitions relating to the regularity lemma andtilings. We then prove an elementary number theoretic result (Theorem 5.2) in Section 5 which willbe a crucial tool in overcoming divisibility barriers during the proof of Theorem 1.9. In Section 6we give an overview of the proof of Theorem 1.9 before proving it in Section 7.2. A discussion on the optimality of degree sequence conditions
In this section we describe various notions concerning when a degree sequence condition is ‘best-possible’ in some sense. In particular, we will explain in what way our results (Theorems 1.7and 1.9) are essentially best-possible, as well as how we may be able to strengthen these theoremsfurther. Some of our discussion draws on the survey [3].First we introduce a few definitions. An integer sequence π = ( d ≤ · · · ≤ d n ) is called graphical if there exists a (simple) graph G that has π as its degree sequence. Given a graph property P , wesay that a graphic sequence π forces P if every graph with degree sequence π satisfies property P .Given a property P (such as containing a Hamilton cycle or perfect H -tiling), the ‘holy-grail’ inthe study of degree sequences is to establish all those graphic sequences π that force P .The following theorem of Chv´atal [4] provides an extremely general condition on degree sequencesthat force a Hamilton cycle. Theorem 2.1 (Chv´atal [4]) . Suppose that the degree sequence of a graph G is d ≤ · · · ≤ d n . If n ≥ and d i ≥ i + 1 or d n − i ≥ n − i for all i < n/ then G is Hamiltonian. ote that Chv´atal’s theorem is best-possible in the following sense: if d ≤ · · · ≤ d n is a degreesequence that does not satisfy the condition in Theorem 2.1 then there exists a non-Hamiltoniangraph G whose degree sequence d ′ ≤ · · · ≤ d ′ n is such that d ′ i ≥ d i for all 1 ≤ i ≤ n . (We willinformally refer to a degree sequence result being best-possible in this way as a Chv´atal-type result.)Crucially note though that Chv´atal’s theorem does not describe all those graphic sequences thatforce a Hamilton cycle. For example, graphs with degree sequence (2 , , , ,
2) must be Hamiltonian(in fact, are themselves simply a 5-cycle), but do not satisfy Chv´atal’s condition. More generally,all 2 k -regular graphs on 4 k + 1 vertices are Hamiltonian [17] yet their degree sequences fail thecondition in Theorem 2.1.2.1. Degree sequence conditions forcing perfect H -tilings. At present, for a given fixedgraph H , it seems out of reach to characterise those graphical degree sequences that force a perfect H -tiling, or obtain a Chv´atal-type result in this setting. Thus, it is natural to seek degree sequenceconditions that force a perfect H -tiling, and are best-possible in some weaker sense. For example,consider the following conjecture: Conjecture 2.2 (Balogh, Kostochka and Treglown [2]) . Let n , r ∈ N such that r divides n . Supposethat G is a graph on n vertices with degree sequence d ≤ . . . ≤ d n such that: ( α ) d i ≥ ( r − n/r + i for all i < n/r ; ( β ) d n/r +1 ≥ ( r − n/r .Then G contains a perfect K r -tiling. Conjecture 2.2 is best-possible in the sense that there are examples (see [2, Section 4]) showingthat one cannot replace ( α ) with d i ≥ ( r − n/r + i − single i or ( β ) with d n/r +1 ≥ ( r − n/r − d n/r +2 ≥ ( r − n/r . That is, there is no room to lower the degree sequence conditionfurther, not even by lowering a single entry by just one. (We will informally refer to a degreesequence result being best-possible in this way as a P´osa-type result.) However, Conjecture 2.2, iftrue, is likely still significantly weaker than a Chv´atal-type result. For example, it is easy to seethat any graph G with degree sequence d ≤ · · · ≤ d n satisfying (i) d ≥ r −
1; (ii) d ≥ n −
2; (iii) d n − r +2 ≥ n − K r -tiling even though the condition in Conjecture 2.2 is violated.One could also ask for a P´osa-type strengthening of the K¨uhn–Osthus theorem (Theorem 1.4) forall graphs H . However, obtaining such a result again seems extremely difficult, not only because(the special case) Conjecture 2.2 is still open, but also in general because the ‘correct’ value of theconstant C ( H ) in Theorem 1.4 is not known.2.2. Extremal examples for Theorems 1.7 and 1.9.
Despite the aforementioned challenges,in this paper we have provided degree sequence conditions that force a perfect H -tiling, and arebest-possible in various ways. The following 3 extremal examples demonstrate this. The first showsthat we cannot significantly lower every term in the degree sequence conditions of Theorems 1.7and 1.9 and still ensure a perfect H -tiling for complete r -partite graphs H . The second shows thatthat the ‘slope’ of the degree sequence in Theorem 1.7 is best possible for so-called bottle graphs.The third demonstrates that for any graph H , to ensure a perfect H -tiling (or even an ‘almost’perfect H -tiling) in a graph G on n vertices we cannot have significantly more than ωn/h verticesthat have degree below (cid:16) − χ cr ( H ) + η (cid:17) n . Extremal Example 1.
The following construction (a simple adaption of [20, Proposition 3.1])demonstrates that for most complete r -partite graphs H , one cannot replace the ηn -term in Theo-rems 1.7 and 1.9 with a o ( √ n )-term. Proposition 2.3.
Let r ≥ and H := K t ,...,t r with t i ≥ (for all ≤ i ≤ r ). Let h := | H | . Set σ ∈ R such that σ ( H ) ≤ σ < h/r and ω := ( h − σ ) / ( r − . Let n ∈ N be sufficiently large so hat √ n is an integer that is divisible by h . Set C := √ n/ h . Then there exists a graph G on n vertices whose degree sequence d ≤ · · · ≤ d n satisfies d i ≥ (cid:18) − ω + σh (cid:19) n + σω i + C for all ≤ i ≤ ωnh but such that G does not contain a perfect H -tiling. Proof.
Let G denote the graph on n vertices consisting of r vertex classes V , . . . , V r with | V | = 1, | V | = ωn/h + 1 + Cr , | V | = ( σ + ω ) n/h − − C and | V i | = ωn/h − C if 4 ≤ i ≤ r and whichcontains the following edges: • All possible edges with an endpoint in V and the other endpoint in V ( G ) \ V . (In particular, G [ V ] is complete.); • All edges with an endpoint in V and the other endpoint in V ( G ) \ V ; • All edges with an endpoint in V i and the other endpoint in V ( G ) \ V i for 4 ≤ i ≤ r ; • There are √ n/ V , each of size ⌊ | V | / √ n ⌋ , ⌈ | V | / √ n ⌉ , which coverall of V .In particular, note that the vertex v ∈ V sends all possible edges to V ( G ) \ V but no edges to V .Let d ≤ · · · ≤ d n denote the degree sequence of G . Notice that every vertex in V i for 3 ≤ i ≤ r has degree at least (1 − ω/h ) n + C . Note that ⌊ | V | / √ n ⌋ ≥ √ n/h = 6 Ch ≥ Cr . Thus, thereare √ n/ V of degree at least(1 − ω/h ) n − − Cr + (6 Cr − ≥ (1 − ω/h ) n + C. The remaining ωn/h + 1 + Cr − √ n/ ≤ ωn/h − √ n/ − V have degree at least(1 − ω/h ) n − Cr ≥ (1 − ω/h ) n − σ √ n/ ω + C. Since d G ( v ) ≥ (cid:0) − ω + σh (cid:1) n + C + σ/ω for the vertex v ∈ V we have that d i ≥ (cid:18) − ω + σh (cid:19) n + σω i + C for all 1 ≤ i ≤ ωnh . Suppose that v ∈ V lies in a copy H ′ of H in G . Then by construction of G , two of the vertexclasses U , U of H ′ must lie entirely in V . By definition of H , H ′ [ U ∪ U ] contains a path oflength 3. However, G [ V ] does not contain a path of length 3, a contradiction. Thus, v does not liein a copy of H and so G does not contain a perfect H -tiling. (cid:3) Extremal Example 2.
We require the following definitions. Let t ∈ N . We will refer to a vertexclass of size t of G as a t -class of G . Set r, σ, ω ∈ N and σ < ω . We define the r -partite bottlegraph B with neck σ and width ω to be the complete r -partite graph with one σ -class and ( r − ω -classes.Let η > B be an r -partite bottle graph with neck σ and width ω . The followingextremal example (adapted from Proposition 3.1 in [7]) G on n vertices demonstrates that The-orem 1.7 is best possible for such graphs B , in the sense that G satisfies the degree sequence ofTheorem 1.7 except for a small linear part that only just fails the degree sequence, but does notcontain a perfect B -tiling. In fact, G does not contain a B -tiling that covers all but at most ηn vertices. Proposition 2.4.
Let η > be fixed and n ∈ N such that < /n ≪ η ≪ . Let B be a bottlegraph with neck σ and width ω , where b := | B | . Additionally assume that b divides n . Then for any ≤ k < ωn/b − ( rb + 1) ηn , there exists a graph G on n vertices whose degree sequence d ≤ . . . ≤ d n satisfies d i ≥ (cid:18) − ω + σb (cid:19) n + σω i + ηn for all i ∈ { , . . . , k − , k + rbηn + 1 , . . . , ωn/b } , i = (cid:18) − ω + σb (cid:19) n + l σω k m + ηn for all k ≤ i ≤ k + rbηn ,but such that G does not contain a B -tiling covering all but at most ηn vertices. Proof.
Let G be the graph on n vertices with r + 1 vertex classes V , . . . , V r +1 where • | V | = σn/b ; • | V | = ωn/b − ηn ; • | V | = . . . = | V r | = ωn/b − ( ηn + 1); • | V r +1 | = ( r − ηn + 1) − V as a , a , . . . , a σn/b . Similarly, label the vertices of V as c , c , . . . , c ωn/b − ηn .The edge set of G is constructed as follows.Firstly, let G have the following edges: • All edges with an endpoint in V and the other endpoint in V ( G ) \ V , in particular G [ V ]is complete; • All edges with an endpoint in V i and the other endpoint in V ( G ) \ ( V ∪ V i ) for 2 ≤ i ≤ r + 1; • All edges with both endpoints in V r +1 , in particular G [ V r +1 ] is complete; • Given any 1 ≤ i ≤ ωn/b − ηn and j ≤ ⌈ σi/ω ⌉ include all edges c i a j .So at the moment G does satisfy the degree sequence in Theorem 1.7; we therefore modify G slightly. For all k ≤ i ≤ k + rbηn and ⌈ σk/ω ⌉ + 1 ≤ j ≤ ⌈ σ ( k + rbηn ) /ω ⌉ delete each edgebetween c i and a j . One can easily check that G satisfies the degree sequence in the statement ofthe proposition. In particular, the vertices of degree (cid:0) − ω + σb (cid:1) n + ⌈ σω k ⌉ + ηn are c k , . . . , c k + rbηn .Define A := { a , . . . , a ⌈ σk/ω ⌉ } and C := { c , . . . , c k + rbηn } . Note that there are no edges between C and V \ A in G . Claim 2.5.
Let T be a B -tiling of G . Then T does not cover at least 3 ηn/ C .Firstly, consider any copy B ′ of B in T that contains at least one vertex in V r +1 . Since C is anindependent set in G , observe that B ′ contains at most ω vertices from C . Thus there are at most ω | V r +1 | = ω ( r − ηn + ω ( r −
2) vertices in C covered by copies of B in T that each contain atleast one vertex in V r +1 .Secondly, consider any copy B ′ of B in T that contains at least one vertex from C and no verticesfrom V r +1 . As before, since C is an independent set in G , we have that B ′ contains at most ω vertices from C . Since there are no edges between C and V \ A in G , B ′ contains at least σ verticesin A .These two observations imply that at most ω ( r − ηn + ω ( r −
2) + ⌈ σk/ω ⌉ ( ω/σ ) < k + (cid:0) b ( r −
1) + (cid:1) ηn vertices in C can be covered by T . Since | C | = k + rbηn , we have that T does not cover at least 3 ηn/ C . Therefore, Claim 2.5 holds. Hence G does not have a B -tiling covering all but at most ηn vertices. (cid:3) Extremal Example 3.
Let H be an h -vertex graph, χ ( H ) =: r , σ := σ ( H ) and ω := ( h − σ ) / ( r − n -vertex graphs G for which all butslightly more than ωn/h vertices have degree above (1 − /χ cr ( H ) + o (1)) n , and the remainingvertices have degree (1 − /χ cr ( H ) − o (1)) n , and yet G does not contain a perfect H -tiling. Thus,this shows that one cannot have significantly more than ωn/h ‘small’ degree vertices in Theorem 1.7. Proposition 2.6.
Let η > be fixed. Let H be a graph with χ ( H ) =: r . Let h := | H | , σ := σ ( H ) and set ω := ( h − σ ) / ( r − . Then there exists a graph G on n vertices whose degree sequence d ≤ . . . ≤ d n satisfies d i = (1 − ω/h − ( r − η ) n = (1 − /χ cr ( H ) − ( r − η ) n for all i ≤ ( ω/h + ( r − η ) n,d i ≥ (1 − ω/h + η ) n = (1 − /χ cr ( H ) + η ) n for all i > ( ω/h + ( r − η ) n, ut such that G does not contain an H -tiling covering all but at most ηn vertices. Proof.
Let G be the complete r -partite graph on n vertices with vertex classes V , . . . , V r suchthat • | V | = σnh − ηn , • | V | = ωnh + ( r − ηn , • | V | = . . . = | V r | = ωnh − ηn .Then G satisfies the degree sequence condition in the proposition. The choice in size of V ensuresthat any H -tiling in G covers at most | V | h/σ < n − ηn vertices, as desired. (cid:3) A possible strengthening of Theorem 1.7.
Whilst Proposition 2.3 demonstrates that wecannot lower every term in the degree sequence condition in Theorem 1.7 by much, perhaps onecan cap the degrees as follows.
Question 2.7.
Can the degree sequence condition in Theorem 1.7 be replaced by d i ≥ min (cid:26)(cid:18) − ω + σh (cid:19) n + σω i + ηn, (cid:18) − χ cr ( H ) (cid:19) n + C (cid:27) for all ≤ i ≤ ωnh where C is a constant dependent only on H ? Note that Theorem 1.8 does not quite imply the K¨uhn–Osthus theorem (Theorem 1.4) due to the ηn -terms. On the other hand, an affirmative answer to Question 2.7, together with an analogous‘capped’ version of Theorem 1.8(ii), would fully imply the upper bound in Theorem 1.4.3. Notation and Definitions
Let G be a graph. We define V ( G ) to be the vertex set of G and E ( G ) to be the edge setof G . Let X ⊆ V ( G ). Then G [ X ] is the graph induced by X on G and has vertex set X andedge set E ( G [ X ]) := { xy ∈ E ( G ) : x, y ∈ X } . We also define G \ X to be the graph withvertex set V ( G ) \ X and edge set E ( G \ X ) := { xy ∈ E ( G ) : x, y ∈ V ( G ) \ X } . For each x ∈ V ( G ), we define the neighbourhood of x in G to be N G ( x ) := { y ∈ V ( G ) : xy ∈ E ( G ) } anddefine d G ( x ) := | N G ( x ) | . We drop the subscript G if it is clear from context which graph we areconsidering. We write d G ( x, X ) for the number of edges in G that x sends to vertices in X . Givena subgraph G ′ ⊆ G , we will write d G ( x, G ′ ) := d G ( x, V ( G ′ )). Let A, B ⊆ V ( G ) be disjoint. Thenwe define e G ( A, B ) := |{ xy ∈ E ( G ) : x ∈ A, y ∈ B }| .Let t ∈ N . We define the blow-up G ( t ) to be the graph constructed by first replacing each vertex x ∈ V ( G ) by a set V x of t vertices and then replacing each edge xy ∈ E ( G ) with the edges of thecomplete bipartite graph with vertex sets V x and V y .We write 0 < a ≪ b ≪ c < a, b, c from right to left.More precisely, there exist non-decreasing functions f : (0 , → (0 ,
1] and g : (0 , → (0 ,
1] suchthat for all a ≤ f ( b ) and b ≤ g ( c ) our calculations and arguments in our proofs are correct. Largerhierarchies are defined similarly. Note that a ≪ b implies that we may assume e.g. a < b or a < b .4. Auxiliary results
The regularity and blow-up lemmas.
The results in this section will be employed in ourproof of Theorem 1.9. First we need the following definitions.
Definition 4.1.
Let G = ( A, B ) be a bipartite graph with vertex classes A and B . We define the density of G to be d G ( A, B ) := e G ( A, B ) | A || B | . et ε >
0. We say that G is ε -regular if for all X ⊆ A and Y ⊆ B with | X | > ε | A | and | Y | > ε | B | we have that | d G ( X, Y ) − d G ( A, B ) | < ε . Definition 4.2.
Given ε > d ∈ [0 ,
1] and G = ( A, B ) a bipartite graph, we say that G is ( ε, d ) -superregular if all sets X ⊆ A and Y ⊆ B with | X | ≥ ε | A | and | Y | ≥ ε | B | satisfy that d ( X, Y ) > d ,that d G ( a ) > d | B | for all a ∈ A and that d G ( b ) > d | A | for all b ∈ B .The following groundbreaking result of Szemer´edi [19] will be instrumental in our proof of The-orem 1.9. Lemma 4.3 (Degree form of Szemer´edi’s Regularity lemma [19]) . Let ε ∈ (0 , and M ′ ∈ N .Then there exist natural numbers M and n such that for any graph G on n ≥ n vertices andany d ∈ (0 , there is a partition of the vertices of G into subsets V , V , . . . , V k and a spanningsubgraph G ′ of G such that the following hold: • M ′ ≤ k ≤ M ; • | V | ≤ εn ; • | V | = . . . = | V k | =: q ; • d G ′ ( x ) > d G ( x ) − ( d + ε ) n for all x ∈ V ( G ) ; • e ( G ′ [ V i ]) = 0 for all i ≥ ; • For all ≤ i, j ≤ k with i = j the pair ( V i , V j ) G ′ is ε -regular and has density either or atleast d . We call V , . . . , V k the clusters of our partition, V the exceptional set and G ′ the pure graph .We define the reduced graph R of G with parameters ε , d and M ′ to be the graph whose vertex setis V , . . . , V k and in which V i V j is an edge if and only if ( V i , V j ) G ′ is ε -regular with density at least d . Note also that | R | = k .We will apply the following well known fact, in conjunction with Lemma 4.5 (below), in Sec-tion 7.2. Fact 4.4.
Let < ε < α and ε ′ := max { ε/α, ε } . Let ( A, B ) be an ε -regular pair of density d .Suppose A ′ ⊆ A and B ′ ⊆ B where | A ′ | ≥ α | A | and | B ′ | ≥ α | B | . Then ( A ′ , B ′ ) is an ε ′ -regular pairwith density d ′ where | d ′ − d | < ε . Lemma 4.5 (Key lemma [13]) . Suppose that < ε < d , that q, t ∈ N and that R is a graph with V ( R ) = { v , . . . , v k } . We construct a graph G as follows: Replace every vertex v i ∈ V ( R ) with aset V i of q vertices and replace each edge of R with an ε -regular pair of density at least d . For each v i ∈ V ( R ) , let U i denote the set of t vertices in R ( t ) corresponding to v i . Let H be a subgraph of R ( t ) with maximum degree ∆ and set h := | H | . Set δ := d − ε and ε := δ ∆ / (2 + ∆) . If ε ≤ ε and t − ≤ ε q then there are at least ( ε q ) h labelled copies of H in G so that if x ∈ V ( H ) lies in U i in R(t), then x is embedded into V i in G . Let G be a graph as in Theorem 1.9 and R a reduced graph of G . The next well known lemmaessentially says that R ‘inherits’ the degree sequence of G . Lemma 4.6 (See e.g. [7]) . Set M ′ , n ∈ N and ε, d, η, b, ω, σ to be positive constants such that /n ≪ /M ′ ≪ ε ≪ d ≪ η, /b and where ω + σ ≤ b . Suppose G is a graph on n ≥ n verticeswith degree sequence d ≤ . . . ≤ d n such that d i ≥ b − ω − σb n + σω i + ηn for all ≤ i ≤ ωnb . et R be the reduced graph of G with parameters ε , M ′ and d and set k := | R | . Then R has degreesequence d R, ≤ . . . ≤ d R,k such that d R,i ≥ b − ω − σb k + σω i + ηk for all ≤ i ≤ ωkb . Let G and H be graphs and R be a reduced graph of G . Let H be a perfect H -tiling in R . Thefollowing result ensures that after removing only a few vertices from each cluster in R every edgein each copy of H ∈ H corresponds to a superregular pair. This will be essential to apply Lemma4.8 in Section 7.4. Proposition 4.7 (See e.g. [16]) . Let G be a graph, ε, d ∈ (0 , and M ′ , ∆ ∈ N . Apply Lemma 4.3to G with parameters ε, M ′ and d to obtain a reduced graph R with clusters of size q . Let H be asubgraph of the reduced graph R with ∆( H ) ≤ ∆ and label the vertices of H as V , . . . , V | H | . Theneach vertex V i of H contains a subset V ′ i of size (1 − ε ∆) q such that for every edge V i V j of H thegraph ( V ′ i , V ′ j ) G ′ is ( ε/ (1 − ε ∆) , d − (1 + ∆) ε ) -superregular. The following fundamental result of Koml´os, S´ark¨ozy and Szemer´edi [11], known as the
Blow-uplemma , essentially says that ( ε, d )-superregular pairs behave like complete bipartite graphs withrespect to containing bounded degree subgraphs.
Lemma 4.8 (Blow-up lemma [11])) . Given a graph F on vertices { , . . . , f } and d, ∆ > , thereexists an ε = ε ( d, ∆ , f ) > such that the following holds. Given L , . . . , L f ∈ N and ε ≤ ε , let F ∗ be the graph obtained from F by replacing each vertex i ∈ F with a set V i of L i new verticesand joining all vertices in V i to all vertices in V j whenever ij is an edge of F . Let G be a spanningsubgraph of F ∗ such that for every edge ij ∈ F the pair ( V i , V j ) G is ( ε, d ) -superregular. Then G contains a copy of every subgraph H of F ∗ with ∆( H ) ≤ ∆ . Tilings in complete graphs.
In [14], the following result of K¨uhn and Osthus is essentialto their proof of Theorem 1.5.
Lemma 4.9. [15, Lemma 12]
Let H be a graph with χ ( H ) =: r ≥ and hcf( H ) = 1 . Let h := | H | and ω ( H ) := ( h − σ ( H )) / ( r − . Let < β ≪ λ ≪ σ ( H ) /ω ( H ) , − σ ( H ) /ω ( H ) , /h be positiveconstants. Suppose that F is a complete r -partite graph with vertex classes U , . . . , U r such that: < / | F | ≪ β ; | F | is divisible by h ; (1 − λ / ) | U r | ≤ σ ( H ) | U i | /ω ( H ) ≤ (1 − λ ) | U r | for all i < r ; || U i | − | U j || ≤ β | F | whenever ≤ i < j < r . Then F contains a perfect H -tiling. We will use the Blow-up lemma in tandem with the following generalisation of Lemma 4.9 toconclude that a particular tiling that we construct in a reduced graph R guarantees a perfect H -tiling in our original graph G . Lemma 4.10.
Let H be a graph with χ ( H ) =: r ≥ and hcf( H ) = 1 . Let h := | H | . Set σ ∈ R such that σ ( H ) ≤ σ < h/r and ω := ( h − σ ) / ( r − . Let < β ≪ λ ≪ σ/ω , − σ/ω , /h bepositive constants. Suppose that F is a complete r -partite graph with vertex classes U , . . . , U r suchthat: < / | F | ≪ β ; | F | is divisible by h ; (1 − λ / ) | U r | ≤ σ | U i | /ω ≤ (1 − λ ) | U r | for all i < r ; || U i | − | U j || ≤ β | F | whenever ≤ i < j < r . Then F contains a perfect H -tiling. Proof.
Note we may assume that σ > σ ( H ) as otherwise the result follows immediately fromLemma 4.9. We choose β ≪ β ≪ λ ≪ λ where β and λ are as in Lemma 4.9. Additionallywe may assume β , λ ≪ ( σ/ω − σ ( H ) /ω ( H )).Let F be as in the statement of the lemma. Set H ∗ to be the complete balanced r -partite graphon rh vertices (that is, each vertex class of H ∗ has size h ). Observe that H ∗ has a perfect H -tilingusing precisely r copies of H .Repeatedly delete disjoint copies of H ∗ from F (and therefore update the classes U , . . . , U r )until the first point for which there is some i < r such that (1 − λ / / | U r | ≤ σ ( H ) | U i | /ω ( H ) ≤ − λ ) | U r | . Call the resulting graph F ′ . Note that σ/ω > σ ( H ) /ω ( H ), so we can indeed obtain F ′ .Further note that the choice of β ensures each class U j still contains at least a β / -proportion ofthe vertices it started with. So now || U i |−| U j || ≤ β | F | ≤ β / | F ′ | ≤ β | F ′ | whenever 1 ≤ i < j < r .Moreover, this implies (1 − λ / ) | U r | ≤ σ ( H ) | U j | /ω ( H ) ≤ (1 − λ ) | U r | for all j < r . Thus, byLemma 4.9, F ′ contains a perfect H -tiling and therefore, so too does F , as desired. (cid:3) A degree sequence Koml´os theorem.
In [14], K¨uhn and Osthus begin their proof ofTheorem 1.5 by applying Koml´os’ theorem (Theorem 1.3). In our proof of Theorem 1.9 we will usethe following degree sequence version of Koml´os’ theorem that the authors and Liu proved in [7].
Theorem 4.11. [7, Theorem 8.1]
Let η > and H be a graph with χ ( H ) = r and h := | H | .Set σ ∈ R such that σ ( H ) ≤ σ ≤ h/r and ω := ( h − σ ) / ( r − . Then there exists an integer n = n ( η, σ, H ) ∈ N such that the following holds. Suppose G is a graph on n ≥ n vertices withdegree sequence d ≤ . . . ≤ d n such that d i ≥ (cid:18) − ω + σh (cid:19) n + σω i for all ≤ i ≤ ωnh .Then G contains an H -tiling covering all but at most ηn vertices. B´ezout’s Lemma.
To prove Theorem 5.2 we will need the following elementary arithmeticresult.
Lemma 4.12 (B´ezout’s Lemma) . Let a , a , . . . , a t ∈ Z . Then there exist integers y , y , . . . , y t ∈ Z such that t X i =1 y i a i = hcf( a , a , . . . , a t ) where hcf( a , a , . . . , a t ) is the highest common factor of a , a , . . . , a t . A tool for the proof of Theorem 1.9
In this section, we prove a theorem (Theorem 5.2) that will be used in Sections 7.3.1 and 7.3.2of the proof of Theorem 1.9. At the beginning of Section 7.3, we will have a certain ˆ B -tilingˆ B of a reduced graph R (the graph ˆ B will be defined later). Denote the copies of ˆ B in ˆ B byˆ B , ˆ B , . . . , ˆ B ˆ k . For applications of Lemma 4.10 required at the end of our proof of Theorem 1.9,we will need | V G ( ˆ B i ) | to be divisible by h for each 1 ≤ i ≤ ˆ k . The following theorem is the crucialtool for ensuring we can remove copies of H from G to achieve this.For a graph H with χ ( H ) = r , recall that C H is the set of all optimal colourings of H and thatgiven an optimal colouring c ∈ C H we let x c, ≤ x c, ≤ . . . ≤ x c,r denote the sizes of the colourclasses of c . We require the following definitions. Definition 5.1.
Let H be a graph with χ ( H ) =: r . Fix ≤ p ≤ r − . For each c ∈ C H , define D c to be the multiset [ x c, , x c, , . . . , x c,r ] . We say that A is a p -subset contained in D c if A is a multiset, | A | = p and A = [ x c,j , x c,j , . . . , x c,j p ] where j , j , . . . , j p ∈ { , . . . , r } are distinct. Let z p := (cid:16) rp (cid:17) bethe number of p -subsets contained in D c . For each colouring c ∈ C H , label the p -subsets containedin D c by A p,c, , A p,c, , . . . , A p,c,z p . Let S p,c,J := P x ∈ A p,c,J x for each c ∈ C H , ≤ J ≤ z p . Theorem 5.2.
Let H be an r -partite graph and let h := | H | . Fix ≤ p ≤ r − . Let b be thenumber of components of H and t , . . . , t b be the sizes of the components of H . Then if r = 2 and hcf c ( H ) = 1 , there exists a collection of non-negative integers { a i : 1 ≤ i ≤ b } and ¯ a ∈ N such that a i ≤ ¯ a for all ≤ i ≤ b, and b X i =1 a i t i ≡ h. • if r ≥ and hcf χ ( H ) = 1 , there exists a collection of non-negative integers { a p,c,i : c ∈ C H , ≤ i ≤ z p } and ¯ a ∈ N such that a p,c,i ≤ ¯ a for all c ∈ C H and ≤ i ≤ z p , and X c ∈ C H z p X i =1 a p,c,i S p,c,i ≡ h. For each 1 ≤ p ≤ r − c ∈ C H and j ∈ { , . . . , r } , let Z p,c,j be the multiset defined by thefollowing table: Colour class size x c, · · · x c,j − x c,j x c,j +1 · · · x c,r Multiplicity in Z p,c,j p · · · p p + 1 p · · · p The following fact will be useful in our proof of Theorem 5.2.
Fact 5.3.
For any ≤ J, L ≤ r , we can partition Z p,c,J into { x c,L } and r p -subsets contained in D c . Proof of Theorem 5.2.
Firstly, we will consider the case when r = 2 and hcf c ( H ) = 1. So H must have multiple components. The sizes of these components of H are t , t , . . . , t b . Sincehcf c ( H ) = 1, by Bezout’s Lemma (Lemma 4.12) there exist integers a ′ , . . . , a ′ b such that b X i =1 a ′ i t i = hcf( t , . . . , t b ) = 1 . Since P bi =1 t i = h , there exists ˆ a ∈ N ∪ { } such that b X i =1 ( a ′ i + ˆ a ) t i ≡ h and a ′ i + ˆ a ≥ ≤ i ≤ b. For each 1 ≤ i ≤ b , take a i := a ′ i + ˆ a and ¯ a := max i =1 ,...,b a i .Next consider when r ≥
3. Instead of explicitly calculating a p,c,i for each c ∈ C H , 1 ≤ i ≤ z p ,we will construct for each c ∈ C H a multiset X c of bounded size which can be partitioned into p -subsets contained in D c . Further, we will construct our multisets X c such that X c ∈ C H X x ∈ X c x ≡ h. Observe that constructing such multisets X c immediately yields a collection of non-negative integers { a p,c,i : c ∈ C H , ≤ i ≤ z p } that satisfy the conditions in Theorem 5.2. Indeed, for each c ∈ C H and 1 ≤ i ≤ z p , we take a p,c,i to be precisely the number of times A p,c,i occurs in the partition of X c into p -subsets. n order to start constructing our multisets X c , we define the following multiset: D ∗ ( H ) := [ c ∈ C H [ x c,j +1 − x c,j | j = 1 , . . . , r − . As hcf χ ( H ) = 1 we can apply Lemma 4.12 to the multiset D ∗ ( H ) to get for each c ∈ C H ,1 ≤ j ≤ r − b c,j such that the following holds:(1) X c ∈ C H r − X j =1 b c,j ( x c,j +1 − x c,j ) ≡ h. We now construct our multisets X c . Fix c ∈ C H . Let t c ∈ N be the smallest t ′ ∈ N such that pt ′ ≥ max {| b c, | , | b c, − b c, | , | b c, − b c, | , . . . , | b c,r − − b c,r − | , | b c,r − |} . Then pt c − b c, , pt c + b c, − b c, , pt c + b c, − b c, , . . . , pt c + b c,r − − b c,r − , pt c + b c,r − are non-negativeintegers. Let Y c be the multiset defined by the following table:Colour class size x c, x c, x c, · · · x c,r − x c,r Multiplicity in Y c pt c − b c, pt c + b c, − b c, pt c + b c, − b c, · · · pt c + b c,r − − b c,r − pt c + b c,r − Then | Y c | = rpt c . If we can partition Y c into p -subsets contained in D c then we take X c := Y c .Assume we cannot. Then the multiplicities of x c, , . . . , x c,r in Y c must be sufficiently different fromone another. We employ the following algorithm which transforms Y c into a multiset which can bepartitioned into p -subsets contained in D c using Fact 5.3. To state the algorithm we require thefollowing definition. Definition 5.4.
For each c ∈ C H , 1 ≤ i ≤ r , let m c,i be the multiplicity of x c,i in Y c . Let J, L ∈ { , . . . , r } such that J = L ; m c,J ≥ P ri =1 m c,i r ; m c,L ≤ P ri =1 m c,i r ; m c,L +1 = m c,J ; m c,L = m c,J .Let Y ′ c := Y c − { x c,J } + { x c,L } . Then we say that Y ′ c is more balanced than Y c . Algorithm. Let Q := ∅ . If | m c,i − m c,j | = 0 for all ≤ i, j ≤ r , output Y c and Q . Otherwise, choose J, L ∈ { , . . . , r } such that Y ′ c := Y c − { x c,J } + { x c,L } is more balanced than Y c . Add p copies of H with colouring c to Y c . That is, x c,i now has multiplicity m c,i + p in Y c for each ≤ i ≤ r . Take Z p,c,J to be the union of { x c,J } and these p copies of H . By Fact 5.3 there exists apartition of Z p,c,J into { x c,L } and r p -subsets contained in D c . Place into Q these r p -subsets contained in D c . Take Y c := Y ′ c and update the value of each m c,i (that is, m c,J has decreased by 1 and m c,L has increased by 1). Go to Step 2. Therefore, at the end of the algorithm | Y c | = rpt c and | m c,i − m c,j | = 0 for all 1 ≤ i, j ≤ r . Inparticular, it is easy to see that Y c now has a partition Q Y c into p -subsets contained in D c . Let ˆ t c be the number of collections of p copies of H added during the algorithm and ˆ t c := t c + ˆ t c . Thenthe multiset ˆ Y c , defined by the table below, can be partitioned into p -subsets contained in D c usingthe partition Q ∪ Q Y c :Colour class size x c, x c, x c, · · · x c,r − x c,r Multiplicity in ˆ Y c p ˆ t c − b c, p ˆ t c + b c, − b c, p ˆ t c + b c, − b c, · · · p ˆ t c + b c,r − − b c,r − p ˆ t c + b c,r − That is, Y ′ c is the multiset Y c except with x c,J having multiplicity m c,J − x c,L having multiplicity m c,L + 1. ake X c := ˆ Y c . We now confirm that our multisets X c satisfy X c ∈ C H X x ∈ X c x ≡ h. By (1) and the definition of X c for each c ∈ C H we have X c ∈ C H X x ∈ X c x = X c ∈ C H r − X j =1 b c,j ( x c,j +1 − x c,j ) + p ˆ t c r X j =1 x c,j = X c ∈ C H r − X j =1 b c,j ( x c,j +1 − x c,j ) + p X c ∈ C H ˆ t c h ( ) ≡ h. Therefore, recalling the discussion earlier in this proof, there must exist the desired collection ofnon-negative integers { a p,c,i : c ∈ C H , ≤ i ≤ z p } . (cid:3) Proof Overview
The rest of this paper will be devoted to proving Theorem 1.9 and here we outline the proof. Asnoted in Section 1.2, our proof follows closely K¨uhn and Osthus’ proof of Theorem 1.5 in [14].Let H , G , η and σ be as in the statement of the theorem. In particular, h := | H | and ω :=( h − σ ) / ( r − σ ∈ Q . First we define abottle graph B that contains a perfect H -tiling. Definition 6.1.
Let a, b ∈ N such that σ = a/b . Let ω ( H ) := ( h − σ ( H )) / ( r − and ˆ c := b ( r − ω ( H ) − σ ( H )) . Define B to be the r -partite bottle graph with neck σ ˆ c and width ω ˆ c (notethat σ ˆ c, ω ˆ c ∈ N ). Observe that | B | = h ˆ c ; σ ( B ) = σ ˆ c ; ω ( B ) = ω ˆ c . Further, χ cr ( B ) = r − σ/ω = h/ω. Since | B | = h ˆ c ; σ ( B ) = σ ˆ c ; ω ( B ) = ω ˆ c , we have that G satisfies the hypothesis of the degreesequence Koml´os theorem (Theorem 4.11) with B , σ ( B ) and ω ( B ) playing the roles of H , σ and ω respectively. That is, G contains an almost perfect B -tiling. In fact, as the reduced graph R of G almost inherits the degree sequence of G , Theorem 4.11 ensures that R contains an almost perfect B -tiling B . Further note that the choice of ˆ c implies that B has a perfect H -tiling consisting of ˆ c copies of H . (A simple proof of this can be found inside the proof of Theorem 8.1 in [7].)Ideally one would like to use B as a framework to build the perfect H -tiling in G . However, asexplained shortly, we need more flexibility in our tiling in R . Therefore, we introduce the following‘modified’ version of B . Definition 6.2.
Let s ∈ N be sufficiently large and λ ∈ R + be sufficiently small. Recall that σ < ω .Let ˆ B be the r -partite bottle graph with neck σ (1 + λ ) s/ω and width s . Moreover, we choose λ and s such that ˆ B contains a perfect B -tiling. Hence ˆ B contains a perfect H -tiling. Note that χ cr ( ˆ B ) = r − σ (1 + λ ) /ω. We have that σ (1 + λ ) /ω < λ and that σ < ω . ince λ is chosen to be small (and so χ cr ( ˆ B ) is very close to χ cr ( B )), one can still apply Theo-rem 4.11 on inputs ˆ B and R . That is, R contains an almost perfect ˆ B -tiling ˆ B . Denote the copiesof ˆ B in ˆ B by ˆ B , ˆ B , . . . , ˆ B ˆ k . By removing a small number of vertices from each cluster in R , we canensure the edges of each ˆ B i correspond to superregular pairs. Let V denote the set of all verticesin G not contained in the clusters lying in the tiling ˆ B .For each 1 ≤ i ≤ ˆ k , let ˆ G i be the r -partite subgraph of G whose j th vertex class is the union ofall those clusters contained in the j th vertex class of ˆ B i , for each 1 ≤ j ≤ r . Let G ∗ i be the complete r -partite graph on the same vertex set as ˆ G i . We introduce the graph ˆ B (rather than just workingwith B ) since ˆ B has the following crucial property: For each 1 ≤ i ≤ ˆ k we can arbitrarily delete asmall number of vertices from G ∗ i (and correspondingly ˆ G i ) and, provided | V ( G ∗ i ) | is now divisibleby h , the resulting graph satisfies the hypothesis of Lemma 4.10. That is, this graph contains aperfect H -tiling. Then the Blow-up lemma (Lemma 4.8) implies that each ˆ G i contains a perfect H -tiling.We make use of this property of ˆ B as follows: In Section 7.2 we remove copies of H from G that cover all vertices in V , as well as a small (possibly zero) number of vertices from each ˆ G i ; callthis H -tiling (formed from these copies of H ) H . Deleting these covered vertices from each ˆ G i , if | V ( ˆ G i ) | (= | V ( G ∗ i ) | ) is still divisible by h for each 1 ≤ i ≤ ˆ k then each ˆ G i now contains a perfect H -tiling (by our argument above). However, for some i , we may have that | V ( ˆ G i ) | is not divisibleby h . So in Section 7.3 we remove a further bounded number of copies of H , forming an H -tiling H , to ensure | V ( ˆ G i ) | (= | V ( G ∗ i ) | ) is divisible by h for each 1 ≤ i ≤ ˆ k . Thus, we now have thateach ˆ G i contains a perfect H -tiling ˆ H i . Combining the tilings H , H , ˆ H , . . . , ˆ H ˆ k yields a perfect H -tiling in G , as desired.Our argument in Section 7.3 will split into two cases, the first being when χ ( H ) ≥ H is bipartite. This is where our proof differs the most from that in [14] as we mustmake use of Theorem 5.2 to find suitable copies of H to ensure each | V ( ˆ G i ) | is divisible by h .7. Proof of Theorem 1.9
Applying the regularity lemma.
Note that it suffices to prove the theorem in the casewhen σ ∈ Q . Let n be sufficiently large and fix constants that satisfy the following hierarchy(2) 0 < /n ≪ /M ′ ≪ ε ≪ d ≪ η ≪ β ≪ α ≪ λ ≪ η, σ/ω, − σ/ω, /h. As discussed in Section 6, we choose s ∈ N sufficiently large and define ˆ B to be the r -partite bottlegraph with neck σ (1 + λ ) s/ω and width s . As before, we choose λ and s such that ˆ B contains aperfect B -tiling, which implies that ˆ B contains a perfect H -tiling. Note again that χ cr ( ˆ B ) = r − σ (1 + λ ) /ω. Moreover, choose η and M ′ such that η ≪ / | ˆ B | and M ′ ≥ n ( η , σ ( ˆ B ) , ˆ B ) , where n is defined as in Theorem 4.11. Let G be an n -vertex graph as in the statement ofTheorem 1.9. Apply Lemma 4.3 with parameters ε , d and M ′ to G to obtain clusters V , . . . , V k andan exceptional set V , where q := | V | = . . . = | V k | and k ≥ M ′ . Let R be the corresponding reducedgraph. Using (2), we may apply Lemma 4.6 with parameters M ′ , n, ε, d, η, h, ω, σ to conclude that R has degree sequence d R, ≤ d R, ≤ . . . ≤ d R,k where(3) d R,i ≥ (cid:18) − ω + σh (cid:19) k + σω i + ηk ≤ i ≤ ωkh . or a graph F , recall that σ ( F ) denotes the size of the smallest possible colour class in any χ ( F )-colouring of F and ω ( F ) := ( | F | − σ ( F )) / ( χ ( F ) − λ ≪ η , we have that(4) d R,i ≥ − ω ( ˆ B ) + σ ( ˆ B ) | ˆ B | ! k + σ ( ˆ B ) ω ( ˆ B ) i for all 1 ≤ i ≤ ω ( ˆ B ) k | ˆ B | .Since | R | = k ≥ M ′ ≥ n ( η , σ ( ˆ B ) , ˆ B ) and (4) holds, we apply Theorem 4.11 to find a ˆ B -tilingˆ B covering all but at most η k vertices in R . Denote the copies of ˆ B in ˆ B by ˆ B , ˆ B , . . . , ˆ B ˆ k . Nowdelete all clusters not contained in some ˆ B i from R and add the vertices in these clusters to V .Therefore now | V | ≤ εn + η n ≤ η n. From now on, we denote by R the subgraph of the reduced graph induced by all the remainingclusters and redefine k := | R | . Since η ≪ η , (3) implies that R has degree sequence d R, ≤ d R, ≤ . . . ≤ d R,k where(5) d R,i ≥ (cid:18) − ω + σh (cid:19) k + σω i + ηk ≤ i ≤ ωkh .For each ˆ B i in ˆ B , let B i be a perfect B -tiling in ˆ B i (recall that earlier we chose s and λ suchthat ˆ B contains a perfect B -tiling). Let B := S B i and observe that B is a perfect B -tiling in R .To aid with calculations we will often work with B instead of ˆ B .Let q ′ := (1 − ε | ˆ B | ) q . By Proposition 4.7, for all 1 ≤ i ≤ ˆ k we can remove ε | ˆ B | q vertices from eachcluster V a belonging to ˆ B i so that each edge V a V b in ˆ B i now corresponds to a (2 ε, d/ V a , V b ) G ′ . Further, all clusters now have size q ′ and for each edge V a V b in ˆ B i the pair ( V a , V b ) G ′ is a 2 ε -regular pair with density at least d/ V and observe that now, since ε ≪ η , / | ˆ B | ,(6) | V | ≤ η n. From now on, we will refer to the subclusters of size q ′ as the clusters of R .By considering a random partition of each cluster V a , and applying a Chernoff bound, one canobtain the following partition of each cluster. Claim 7.1.
Let V a be a cluster. Then there exists a partition of V a into a red part V reda and a bluepart V bluea such that || V reda | − | V bluea || ≤ εq ′ and || N G ( x ) ∩ V reda | − | N G ( x ) ∩ V bluea || < εq ′ for all x ∈ V ( G ) . Apply Claim 7.1 to every cluster to yield a partition of V ( G ) − V into red and blue vertices. Inthe next section, we will remove vertices of particular copies of H in G from their respective clustersand do so in such a way that we avoid all the red vertices of G . After removing these vertices, wewill be able to conclude that that each (modified) pair ( V a , V b ) G ′ is (5 ε, d/ since V reda and V redb will have had no vertices removed from them. After the next section, we will onlyremove a bounded number of vertices from the clusters, which will not affect the superregularityof pairs of clusters in any significant way. Where V a V b is any edge in any ˆ B i in ˆ B . .2. Covering the exceptional vertices.
As in [14], given x ∈ V , we call a copy of B ∈ B usefulfor x if there exist r − B , each belonging to a different vertex class of B , such that x has at least αq ′ neighbours in each cluster. Denote by k x the number of copies of B in B that areuseful for x . The following calculation demonstrates that k x βq ′ ≥ | V | . By (2) and (6), we have that k x | B | q ′ + ( |B| − k x )( | B | q ′ − (1 − α ) q ′ ˆ c ( ω + σ )) ≥ d G ( x ) − | V |≥ (cid:18) − ω + σh + η (cid:19) q ′ | B ||B| , which implies ( |B| − k x )( − (1 − α ) q ′ ˆ c ( ω + σ )) ≥ (cid:18) − ω + σh + η (cid:19) q ′ h ˆ c |B| . Rearranging, we get k x ≥ |B| (cid:16) hη − α ( ω + σ ) (cid:17) ( ω + σ )(1 − α ) . Since α ≪ η , we have that k x ≥ η |B| . Now as |B| q ′ ≥ n | B | and η ≪ β, η, /h we have that k x βq ′ ≥ η |B| βq ′ / > η n ≥ | V | . Hence we can assign greedily each vertex x ∈ V to a copy B x that is useful for x and do so in sucha way that at most βq ′ vertices in V are assigned to the same copy B ∈ B . Then for each copy B x ∈ B that is useful for some x ∈ V we can apply Lemma 4.5 to find a copy of H containing x which contains no red vertices. We do this as follows:For each x , since ε ≪ α and x has at least αq ′ neighbours in r − B x , Claim 7.1 implies that x has at least αq ′ / r − αq ′ / B x that does not necessarily contain any neighbours of x . Denote this vertex class of B x by C x . Then it is easy to see that we can find subclusters S , . . . , S r of r clusters in B x such that: allvertices in S ∪ . . . ∪ S r are blue vertices; | S i | = αq ′ / i ; every vertex in S ∪ . . . ∪ S r − is aneighbour of x in G . By Fact 4.4, each pair ( S i , S j ), 1 ≤ i < j ≤ r , corresponds to an (8 ε/α )-regularpair in G ′ with density at least d/
3. Using Lemma 4.5 with parameters 8 ε/α, d/ , αq ′ / , h −
1, wefind a copy of H containing x . Since each B ∈ B has been assigned to at most βq ′ vertices in V and β ≪ α (from (2)), we may repeat the above argument to find copies of H that contain eachexceptional vertex in such a way that the copies are disjoint and contain no red vertices. Denotethe H -tiling induced by these copies of H by H . Remove all the vertices lying in these copies of H from their respective clusters. Observe that currently(1 − βh ) q ′ ≤ | V i | ≤ q ′ for each i . .3. Making the blow-up of each B ∈ B divisible by h . For a subgraph S ⊆ R , let V G ( S )denote the union of the clusters in S . We aim to apply Lemma 4.8 to each ˆ B i in ˆ B to find an H -tiling that covers every vertex of V G ( ˆ B i ). Combining these H -tilings with H will result in aperfect H -tiling in G as desired. Recall that, for each 1 ≤ i ≤ ˆ k , ˆ G i is the r -partite subgraph of G ′ whose j th vertex class is the union of all those clusters contained in the j th vertex class of ˆ B i , foreach 1 ≤ j ≤ r . Further, recall that G ∗ i is the complete r -partite graph on the same vertex set asˆ G i . To apply Lemma 4.8 to each ˆ B i in ˆ B we require that each G ∗ i contains a perfect H -tiling. Toguarantee the existence of these perfect H -tilings we will apply Lemma 4.10. To use Lemma 4.10on G ∗ i we require that | V ( ˆ G ∗ i ) | is divisible by h . When we first chose our ˆ B -tiling this was the case.Indeed, as each ˆ B i contained a perfect H -tiling and every cluster V i was the same size, | V ( G ∗ i ) | wasdivisible by h . However, in the last section we took out vertices from G in a greedy way, changingthe sizes of the clusters in R . Hence we cannot guarantee that | V ( G ∗ i ) | is still divisible by h foreach i . Now we will take out a further bounded number of copies of H in G to ensure | V ( G ∗ i ) | isdivisible by h for each 1 ≤ i ≤ ˆ k . In fact, we will ensure | V G ( B ) | is divisible by h for each B ∈ B .We now split into two cases: when r ≥ r = 2. When r ≥ χ ( H ) = 1and this property will be central to our argument. For r = 2, we have that hcf c ( H ) = 1 andhcf χ ( H ) ≤
2. The former property will provide us an easy way of removing copies of H from V ( G ) to ensure | V G ( B ) | is divisible by h for each B ∈ B . Further, we will not need to use theproperty that hcf χ ( H ) ≤ χ ( H ) ≤ Case 1: r ≥ . To assist in our argument, we define an auxiliary graph F whose vertices arethe copies of B in B and for B , B ∈ V ( F ), we let B B be an edge in F if and only if there existsa vertex x in V R ( B ) and r − V R ( B ) (or vice versa) such that these r vertices inducea K r in R . Assume F is connected and let B B be an edge in F . Then we may apply Lemma 4.5to find h − H which each have one vertex in V G ( B ) and all other vertices in V G ( B ) (or vice versa). This means that we can remove at most h − H to ensure V G ( B )is divisble by h . Continuing in this way we can ‘shift the remainders mod h ’ along a spanning treeof F to ensure | V G ( B ) | is divisible by h for each B ∈ B . (Indeed, since n is divisible by h we havethat P B ∈B | V G ( B ) | is divisible by h .)So assume F is not connected. Let C be the set of all components of F . For C ∈ C we will write V R ( C ) for the set of clusters in R belonging to copies of B in C and V G ( C ) for the union of saidclusters. In what follows our aim is to remove a bounded number of copies of H to ensure that foreach component C ∈ C we have that | V G ( C ) | is divisible by h . Then we can apply our previousargument to spanning trees of each component to achieve that | V G ( B ) | is divisible by h for each B ∈ B .Call vertices in R of degree at least(7) (1 − ω/h + η/ k big . If a vertex is not big, call it small . Note by (5) that all but at most ωk/h − R arebig. Claim 7.2.
Let C , C ∈ C , C = C and let a ∈ V R ( C ) . Then | N R ( a ) ∩ V R ( C ) | < (cid:18) − ω + σh + η (cid:19) | V R ( C ) | . Proof.
Recall that B has width ω ˆ c . Suppose Claim 7.2 is false. Then there exists some B ∈ B such that B ∈ C and | N R ( a ) ∩ B | ≥ (cid:18) − ω + σh + η (cid:19) | B | = ( r − ω ˆ c + ηh ˆ c . hus a must have neighbours in at least r − B . We can therefore construct acopy of K r in R which consists of a together with r − B . But by definition ofthe auxiliary graph F , we must have that B is adjacent in F to the copy of B in B that contains a . This contradicts that C and C were different components of F . Thus Claim 7.2 holds. (cid:3) Claim 7.3.
There exist components C , C ∈ C , C = C , a big vertex x ∈ V ( R ) and another (notnecessarily big) vertex x ∈ V ( R ) such that x ∈ V ( C ) , x ∈ V ( C ) and x x ∈ E ( R ) . Proof.
Take some big vertex x ∈ V ( R ). Then x is in V R ( C x ) for some component C x of F .Assume | C x | ≥ (1 − ω/h + η/ k , as otherwise x has a neighbour in R outside of C x and we aredone. Since r ≥ | R \ V R ( C x ) | ≤ ( ω/h − η/ k < (1 − ω/h + η/ k. If R \ V R ( C x ) contains any big vertex y , then y has a neighbour in V R ( C x ) since | R \ V R ( C x ) | < (1 − ω/h + η/ k and we are done. Hence assume all big vertices are in V R ( C x ). Then all verticesin R \ V R ( C x ) are small vertices. Let z be a small vertex in R \ V R ( C x ). Since r ≥ d R ( z ) ≥ (1 − ( ω + σ ) /h + η/ k ≥ ( ω/h + η/ k. Since there are at most ωk/h − R , we have that z has a neighbour w which is abig vertex. But then w ∈ V R ( C x ). Thus Claim 7.3 holds. (cid:3) Claim 7.4.
There exists a copy K ′ of K r in R which has vertices in at least two components of F . Proof.
By Claim 7.3, there exist components C , C ∈ C , a big vertex x ∈ V ( R ) and anothervertex x ∈ V ( R ) such that x ∈ V R ( C ), x ∈ V R ( C ) and x x ∈ E ( R ). By (5) and (7), x and x have a common neighbourhood of size at least(( r − ω/h + η/ k. If r = 3, then we choose x in the common neighbourhood of x and x , and we are done. Soassume r ≥
4. Since there are at most ωk/h small vertices, we can choose a big vertex x in thecommon neighbourhood of x and x . Then x , x and x have a common neighbourhood of sizeat least (( r − ω/h + 3 η/ k. If r = 4, then we choose x in the common neighbourhood of x , x and x and we are done.Otherwise r ≥ (cid:3) For such a copy K ′ of K r in R , we now show that we can take out a bounded number of copiesof H from the clusters corresponding to the vertices of K ′ in such a way that that leaves one of thecomponents C ∈ C with | V G ( C ) | divisible by h . We use Theorem 5.2 and Lemma 4.5 to achievethis. We will then repeat this process to ensure | V G ( B ) | is divisible by h for each B ∈ B . Claim 7.5.
There exists t ∈ N such that by removing at most t + ( |B| − |C| )( h − copies of H from G we can ensure | V G ( B ) | is divisible by h for each B ∈ B . Proof.
Firstly, for each component C ∈ C we will remove copies of H to ensure | V G ( C ) | isdivisible by h . Apply Claim 7.4 to find a copy K ′ of K r in R which has vertices in at least twocomponents of F . Let C ∗ be a component of F which contains at least one vertex of K ′ . Let p bethe number of vertices of K ′ contained in C ∗ and observe that 1 ≤ p ≤ r −
1. Let 0 ≤ g ≤ h − | V G ( C ∗ ) | ≡ g mod h . If g = 0 then | V G ( C ∗ ) | is divisible by h and we consider thegraphs F := F − V ( C ∗ ) and R := R − V R ( C ∗ ). So assume 1 ≤ g ≤ h −
1. Observe that we canapply Lemma 4.5 to find any bounded number of disjoint copies of H in G in the clusters of K ′ (see the end of Section 7.2). For any copy H ′ of H in G in the clusters of K ′ there are precisely p colour classes of some colouring c of H ′ contained in the clusters of K ′ in V G ( C ∗ ). Moreover, iven any colouring c of H and p -subset P contained in D c we can find any bounded number ofdisjoint copies H ′ of H in G with colouring c in the clusters of K ′ so that the colour classes of H ′ in V G ( C ∗ ) correspond to the p -subset P . Moreover there exists j ∈ { , . . . , z p } such that P = A p,c,j (recall this notation from Definition 5.1). Thus removing such a copy H ′ of H from G wouldresult in removing precisely S p,c,j vertices from V G ( C ∗ ). By Theorem 5.2, there exist a collectionof non-negative integers { a p,c,i : c ∈ C H , ≤ i ≤ z p } and ¯ a ∈ N such that a p,c,i ≤ ¯ a for all c ∈ C H , ≤ i ≤ z p , and g · X c ∈ C H z p X i =1 a p,c,i S p,c,i ≡ g mod h. Hence we can remove g · X c ∈ C H z p X i =1 a p,c,i ≤ ( h − a | C H | z p suitable disjoint copies of H in G in the clusters of K ′ to make | V G ( C ∗ ) | divisible by h .Next we consider graphs F := F − V ( C ∗ ) and R := R − V R ( C ∗ ). Let k := | R | . Claim 7.2and (5) together give us that R has degree sequence d R , ≤ . . . ≤ d R ,k where d R ,i ≥ (cid:18) − ω + σh (cid:19) k + σω i + ηk ≤ i ≤ ωk h .Suppose |C| ≥
3. Arguing as in Claims 7.3 and 7.4 we can find a copy K ′ of K r in R which hasvertices in at least two components of F . Let C ∗∗ be a component of F which contains at leastone vertex of K ′ . As before by removing at most ( h − a | C H | z p copies of H from the clustersof K ′ we can make | V G ( C ∗ ) | divisible by h . Since | G | is divisible by h , we can continue in thisway to make | V G ( C ) | divisible by h for each component C ∈ C . We then apply the ‘shifting theremainders mod h ’ argument mentioned earlier during the ‘ F connected’ case to guarantee that | B | is divisible by h for each B ∈ B . In this process we removed at most ( |C| − h − a | C H | z p disjoint copies of H from G . Each time we use the ‘shifting the remainders mod h ’ argument on aconnected component C ∈ C we remove at most ( | C | − h −
1) disjoint copies of H in G . Henceoverall we remove at most ( |C| − h − a | C H | z p + ( |B| − |C| )( h −
1) disjoint copies of H in G .Denote this H -tiling (formed from these copies of H ) by H . (cid:3) Observe that now (1 − hβ ) q ′ ≤ | V i | ≤ q ′ for each i since we only removed a bounded number of vertices from G .7.3.2. Case 2: r = 2 . As in the statement of Theorem 5.2, let b be the number of components of H and t , . . . , t b be the sizes of the components of H . By Theorem 5.2, there exists a collection ofnon-negative integers { a i : 1 ≤ i ≤ b } and ¯ a ∈ N such that a i ≤ ¯ a for all 1 ≤ i ≤ b, and b X i =1 a i t i ≡ h. Let B , B ∈ B . If | V G ( B ) | ≡ h , define B := B \ B . If not, let p ∈ { , . . . , h − } suchthat | V G ( B ) | ≡ p mod h . Remove p P bi =1 a i copies of H from V G ( B ) ∪ V G ( B ) in the followingway: For each 1 ≤ i ≤ b , remove pa i copies of H from V G ( B ) ∪ V G ( B ) such that the component f order t i is in V G ( B ) and all other components are in V G ( B ). Since p P bi =1 a i t i ≡ p mod h , byremoving these p P bi =1 a i copies of H from V G ( B ) ∪ V G ( B ) we now have that | V G ( B ) | is divisibleby h . Define B := B \ B .Let B ′ , B ′ ∈ B . If | V G ( B ′ ) | ≡ h , define B := B \ B ′ . If not, let p ′ ∈ { , . . . , h − } such that | V G ( B ′ ) | ≡ p ′ mod h . Remove p ′ P bi =1 a i copies of H from V G ( B ′ ) ∪ V G ( B ′ ) in the sameway as before. Define B := B \ B ′ . Continuing in the same way, we see that by removing at most(8) ( |B| − h − b ¯ a copies of H we can ensure that | B | is divisible by h for each B ∈ B . Denote this H -tiling (formedfrom these copies of H ) by H .Observe that now (1 − hβ ) q ′ ≤ | V i | ≤ q ′ for each i since we only removed a bounded number of vertices.7.4. Completing the perfect tiling.
As we related at the beginning of Section 7.3, we aim toapply Lemma 4.8 to each ˆ B i ⊆ R (1 ≤ i ≤ ˆ k ) where the vertices of R are the now modified clusters– modified by the removing of copies of H in previous sections. Recall that, for each 1 ≤ i ≤ ˆ k , ˆ G i is the r -partite subgraph of G ′ whose j th vertex class is the union of all those clusters contained inthe j th vertex class of ˆ B i , for each 1 ≤ j ≤ r . Observe that in Section 7.3 we made | ˆ G i | = | V G ( ˆ B i ) | divisible by h for each i . Further, (1 − hβ ) q ′ ≤ | V i | ≤ q ′ for each i . Recall that G ∗ i is the complete r -partite graph on the same vertex set as ˆ G i . Since0 < hβ ≪ σ/ω, − σ/ω, /h by (2), we can apply Lemma 4.10 to conclude that each G ∗ i containsa perfect H -tiling.Furthermore, pairs of clusters that correspond to edges of ˆ B i are still (6 ε, d/ H which avoided red vertices, resulting in each pairof clusters (in a copy of H ) being (5 ε, d/ r = 2, we removed only a constant number of vertices from each cluster. Hence each pair ofclusters (in a copy of H ) is (6 ε, d/ H -tiling ˆ H i in ˆ G i for each1 ≤ i ≤ ˆ k . Then H ∪ H ∪ ˆ H ∪ . . . ∪ ˆ H ˆ k is a perfect H -tiling in G . Hence we have proved Theorem 1.9.8. Acknowledgements
The first author would like to thank Pat Devlin for a helpful conversation at Building BridgesII.
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