University of Birmingham A degree sequence version of the Kühn-Osthus tiling theorem

. A fundamental result of K¨uhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H -tiling. We prove a degree sequence version of this result which allows for a signiﬁcant number of vertices to have lower degree.

1. Introduction 1.1.Minimum degree conditions forcing tilings.A substantial branch of extremal graph theory concerns the study of tilings.Given two graphs H and G, an H-tiling in G is a collection of 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 whether a graph contains a perfect H-tiling is NP-complete [6].Thus, there has been a focus on establishing sufficient conditions to force a perfect H-tiling.The seminal Hajnal-Szemerédi theorem [5] characterises the minimum degree that ensures a graph contains a perfect K r -tiling.
Theorem 1.1 (Hajnal and Szemerédi [5]).Every graph G whose order n is divisible by r and whose minimum degree satisfies δ(G) ≥ (1 − 1/r)n contains a perfect K r -tiling.Moreover, there are n-vertex graphs G with δ(G) = (1 − 1/r)n − 1 that do not contain a perfect K r -tiling.
The following result of Alon and Yuster [1] shows that any sufficiently large graph G with minimum degree slightly above that in Theorem 1.1 in fact contains a perfect H-tiling for any graph H with χ(H) = r.Theorem 1.2 (Alon and Yuster [1]).Suppose that γ > 0 and H is a graph with χ(H) = r.Then there exists an integer n 0 = n 0 (γ, H) such that the following holds.If G is a graph whose order n ≥ n 0 is divisible by |H| and δ(G) ≥ (1 − 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 the term γn.Indeed, for many graphs H there are so-called divisibility barrier constructions G on n vertices that have minimum degree (1 − 1/χ(H))n − 1 but fail to contain a perfect H-tiling (see [15,Section 2]).However, Komlós, Sárközy and Szemerédi [12] proved that the term γn in Theorem 1.2 can be replaced with a constant dependent only on H. Further, as discussed shortly, Kühn and Osthus [14,15] proved that there are also many graphs H for which one can significantly reduce the minimum degree condition in Theorem 1.2.
In a related direction, Komlós [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, jfh337@bham.ac.uk.AT: University of Birmingham, United Kingdom, a.c.treglown@bham.ac.uk.
of the Alon-Yuster theorem by the so-called critical chromatic number χ cr (H) of H.Here χ cr (H) is defined as where σ(H) denotes the size of the smallest possible colour class in any χ(H)-colouring of H.Note that all graphs H satisfy χ(H) − 1 < 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 (Komlós [10]).Let η > 0 and let H be a graph.Then there exists an integer n 0 = n 0 (η, H) such that every graph G on n ≥ n 0 vertices with 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 one cannot replace the (1 − 1/χ cr (H)) term here with any smaller fixed constant (this is a consequence of [10,Theorem 7]).Further, for any x ∈ (0, 1) and sufficiently large n, Komlós [10] determined the minimum degree threshold that ensures a graph G on n vertices contains an H-tiling covering at least xn vertices.Shoukoufandeh and Zhao [18] later proved that the number of uncovered vertices in Theorem 1.3 can be reduced to a constant dependent only on H.
Kühn and Osthus [14,15] showed that for many graphs H, a minimum degree slightly above that in Komlós' theorem actually ensures a perfect H-tiling.To state their result we need to introduce some 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 We denote by hcf χ (H) the highest common factor of all integers in D(H).If D(H) = {0} then we define hcf χ (H) := ∞.We write hcf c (H) for the highest common factor of all the orders of components of H.For non-bipartite graphs H we say that hcf(H) = 1 if hcf χ (H) = 1.If χ(H) = 2 then we say hcf(H) = 1 if hcf c (H) = 1 and hcf χ (H) ≤ 2. (See [15] for some examples.)Set Also let δ(H, n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-tiling.
When hcf(H) = 1, Kühn and Osthus showed that χ cr (H) is the parameter governing the minimum degree condition that ensures a perfect H-tiling.When hcf(H) = 1, χ(H) instead is the relevant parameter.
Theorem 1.4 (Kühn and Osthus [15]).For every graph H there exists a constant C = C(H) such that Intuitively speaking, graphs H with hcf(H) = 1 avoid certain divisibility barrier problems when seeking a perfect H-tiling, thus ensuring the lower threshold in this case in Theorem 1.4.Earlier Kühn and Osthus [14] had proven a version of Theorem 1.4 for graphs H with χ(H) ≥ 3 and hcf(H) = 1; there though the constant C(H) was replaced with a linear error term.We now state this result explicitly for future reference.
Theorem 1.5 (Kühn and Osthus [14]).Let η > 0 and H be a graph with hcf χ (H) = 1 and χ(H) =: r ≥ 3. Then there exists an integer n 0 = n 0 (η, H) such that the following holds.Let G be a graph on n ≥ n 0 vertices such that |H| divides n and Then G contains a perfect H-tiling.
1.2.Degree sequence conditions forcing tilings.As discussed in the previous subsection, the minimum degree conditions in each of the Hajnal-Szemerédi theorem, Komlós' theorem and the Kühn-Osthus theorem are essentially best-possible.However, this does not mean one cannot seek significant 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 with the condition that the sum of the degrees of every pair of non-adjacent vertices in G is at least The focus of this paper concerns degree sequence conditions that force a perfect H-tiling.The study of degree sequence results for tilings was initiated in [2].In particular, a conjecture on a degree sequence strengthening of the Hajnal-Szemerédi theorem was raised [2, Conjecture 7], as well as a degree sequence version of the Alon-Yuster theorem [2, Conjecture 8].In [20] the second author proved the latter conjecture (also yielding an asymptotic version of Conjecture 7 from [2]).
Theorem 1.6 (Treglown [20]).Suppose that η > 0 and H is a graph with χ(H) =: r ≥ 2. Then there exists an integer n 0 = n 0 (η, H) such that the following holds.If G is a graph whose order n ≥ n 0 is divisible by |H|, and whose degree sequence Theorem 1.6 is a significant strengthening of the Alon-Yuster theorem as it allows for n/r vertices to have degree (significantly) below that required in the latter.Further Theorem 1.6 provides the first piece of a degree sequence analogue of the Kühn-Osthus theorem.
The main result in this paper deals with the remaining part of this problem, providing a degree sequence condition that forces a perfect H-tiling for graphs with hcf(H) = 1.
Theorem 1.7.Let η > 0 and H be a graph with hcf(H) = 1 and χ(H) =: r ≥ 2. Let σ := σ(H), h := |H| and ω := (h − σ) /(r − 1).Then there exists an integer n 1 = n 1 (η, H) such that the following holds.Let G be a graph on n ≥ n 1 vertices such that h divides n and G has degree sequence d 1 ≤ . . .≤ d n such that Note that here ω is the average size of the colour classes of H after excluding one of minimal size σ(H).Observe that when i = ωn/h, we have Hence, 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) ≥ 3 and hcf 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 that we cannot replace the ηn-term with a o( √ n)-term (see Proposition 3.3).Theorem 1.7 is also best possible 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 3.6).Thus, it is not possible to allow significantly more 'small' degree vertices in Theorem 1.7.Extremal examples are discussed in more detail in Section 3.2.
Combining Theorem 1.7 with Theorem 1.6 we obtain the following degree sequence version of the Kühn-Osthus theorem (Theorem 1.4).
(i) hcf(H) = 1 and G has degree sequence d 1 ≤ . . .≤ d n such that One can in fact obtain the following generalisation of Theorem 1.7 in which we assume σ(H) ≤ σ < h/r.Theorem 1.9.Let η > 0 and H be a graph with hcf(H) = 1 and χ(H) =: r ≥ 2. Let h := |H|.Set σ ∈ R such that σ(H) ≤ σ < h/r and ω := (h − σ) /(r − 1).Then there exists an integer n 1 = n 1 (η, H) such that the following holds.Let G be a graph on n ≥ n 1 vertices such that h divides n and G has degree sequence d 1 ≤ . . .≤ d n such that Observe that for graphs H with hcf(H) = 1, Theorem 1.9 interpolates between Theorems 1.6 and 1.7.In Section 7, we will prove Theorem 1.9 directly.The proof of Theorem 1.9 follows that of Theorem 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 (see Theorem 5.2).We also make use of a recent degree sequence strengthening of Komlós' theorem proved by the authors and Liu [7].
Since the choice of σ ∈ [σ(H), h/r) is arbitrary, note that Theorem 1.9 provides an infinite collection of degree sequences that force a perfect H-tiling.Having a higher value of σ lowers the starting point of the degree sequence condition, but at the price of a steeper 'slope' and higher value of d ωn/h (see Figure 1).As with Theorem 1.7, for many graphs H, each of these degree sequences is best-possible in the sense that we cannot replace the ηn-term with a o( √ n)-term (see Section 3.2).Note too that we cannot extend Theorem 1.9 to the case when σ < σ(H).Indeed, in this case, if we set η 1 then the degree sequence condition in Theorem 1.9 would allow all vertices in G to have degree below (1 − 1/χ cr (H))n − 1; however, we know from Theorem 1.4 that there 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 introduce some notation and definitions.In Section 3 we discuss various senses of optimality for degree sequence conditions before giving several extremal examples for Theorems 1.7 and 1.9.We also ask whether one can improve Theorem 1.7 by suitably 'capping' the bounds on the degrees of the vertices (see Question 3.7).In Section 4 we give a number of auxiliary results and definitions relating to the regularity lemma and tilings.
We then prove an elementary number theoretic result (Theorem 5.2) in Section 5 which will be a crucial tool in overcoming divisibility barriers during the proof of Theorem 1.9.In Section 6 we give an overview of the proof of Theorem 1.9 before proving it in Section 7.

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 set of G. Let X ⊆ V (G).Then G[X] is the graph induced by X on G and has vertex set X and edge set E(G[X]) := {xy ∈ E(G) : x, y ∈ X}.We also define G \ X to be the graph with vertex set V (G) \ X and edge set E(G \ X) We drop the subscript G if it is clear from context which graph we are considering.We write d G (x, X) for the number of edges in G that x sends to vertices in X.Given a subgraph G ⊆ G, we will write 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 the complete bipartite graph with vertex sets V x and V y .
We write 0 < a b c < 1 to mean that we can choose the constants a, b, c from right to left.More precisely, there exist non-decreasing functions f : (0, 1] → (0, 1] and g : (0, 1] → (0, 1] such that for all a ≤ f (b) and b ≤ g(c) our calculations and arguments in our proofs are correct.Larger hierarchies are defined similarly.Note that a b implies that we may assume e.g. a < b or a < b 2 .

3.
A discussion on the optimality of degree sequence conditions In this section we describe various notions concerning when a degree sequence condition is 'bestpossible' in some sense.In particular, we will explain in what way our results (Theorems 1.7 and 1.9) are essentially best-possible, as well as how we may be able to strengthen these theorems further.Some of our discussion draws on the survey [3].
First we introduce a few definitions.An integer sequence π = (d is called graphic if there exists a (simple) graph G that has π as its degree sequence.Given a graph property P , we say 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' in the study of degree sequences is to establish all those graphic sequences π that force P .
The following theorem of Chvátal [4] provides an extremely general condition on degree sequences that force a Hamilton cycle.Theorem 3.1 (Chvátal [4]).Suppose that the degree sequence of a graph G is Note that Chvátal's theorem is best-possible in the following sense: if d 1 ≤ • • • ≤ d n is a degree sequence that does not satisfy the condition in Theorem 3.1 then there exists a non-Hamiltonian graph G whose degree sequence (We will informally refer to a degree sequence result being best-possible in this way as a Chvátal-type result.) Crucially note though that Chvátal's theorem does not describe all those graphic sequences that force a Hamilton cycle.For example, graphs with degree sequence (2, 2, 2, 2, 2) must be Hamiltonian (in fact, are themselves simply a 5-cycle), but do not satisfy Chvátal's condition.More generally, all 2k-regular graphs on 4k + 1 vertices are Hamiltonian [17] yet their degree sequences fail the condition in Theorem 3.1.
3.1.Degree sequence conditions forcing perfect H-tilings.At present, for a given fixed graph H, it seems out of reach to characterise those graphic degree sequences that force a perfect H-tiling, or obtain a Chvátal-type result in this setting.Thus, it is natural to seek degree sequence conditions that force a perfect H-tiling, and are best-possible in some weaker sense.For example, consider the following conjecture: Conjecture 3.2 (Balogh, Kostochka and Treglown [2]).Let n, r ∈ N such that r divides n.Suppose that G is a graph on n vertices with degree sequence Conjecture 3.2 is best-possible in the sense that there are examples (see [2,Section 4]) showing that one cannot replace (α) with d i ≥ (r − 2)n/r + i − 1 for even a single i or (β) with d n/r+1 ≥ (r − 1)n/r − 1 and d n/r+2 ≥ (r − 1)n/r.That is, there is no room to lower the degree sequence condition further, not even by lowering a single entry by just one.(We will informally refer to a degree sequence result being best-possible in this way as a Pósa-type result.)However, Conjecture 3.2, if true, is likely still significantly weaker than a Chvátal-type result.For example, it is easy to see that any graph G with degree sequence One could also ask for a Pósa-type strengthening of the Kühn-Osthus theorem (Theorem 1.4) for all graphs H.However, obtaining such a result again seems extremely difficult, not only because (the special case) Conjecture 3.2 is still open, but also in general because the 'correct' value of the constant C(H) in Theorem 1.4 is not known.

3.2.
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 are best-possible in various ways.The following 3 extremal examples demonstrate this.The first shows that we cannot significantly lower every term in the degree sequence conditions of Theorems 1.7 and 1.9 and still ensure a perfect H-tiling for complete r-partite graphs H.The second shows that that 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 vertices that have degree below 1 − 1 χcr(H) + η 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 Theorems 1.7 and 1.9 with a o( √ n)-term.
Then there exists a graph G on n vertices whose degree sequence Proof.Let G denote the graph on n vertices consisting of and which contains the following edges: • All possible edges with an endpoint in V 3 and the other endpoint in V (G)\V 1 .(In particular, G[V 3 ] is complete.);• All edges with an endpoint in V 2 and the other endpoint in V (G) \ V 2 ; • All edges with an endpoint in V i and the other endpoint in

The remaining ωn
does not contain a path of length 3, a contradiction.Thus, v does not lie in a copy of H and so G does not contain a perfect H-tiling.
Extremal Example 2. We require the following definitions.Let t ∈ N. We will refer to a vertex class of size t of G as a t-class of G. Set r, σ, ω ∈ N and σ < ω.We define the r-partite bottle graph B with neck σ and width ω to be the complete r-partite graph with one σ-class and (r − 1) ω-classes.
Let η > 0 be fixed.Let B be an r-partite bottle graph with neck σ and width ω.The following extremal example (adapted from Proposition 3.1 in [7]) G on n vertices demonstrates that Theorem 1.7 is best possible for such graphs B, in the sense that G satisfies the degree sequence of Theorem 1.7 except for a small linear part that only just fails the degree sequence, but does not contain a perfect B-tiling.In fact, G does not contain a B-tiling that covers all but at most ηn vertices.
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 1 , . . ., V r+1 where The edge set of G is constructed through the following process.
Initially, let G have the following edges: • All edges with an endpoint in V 1 and the other endpoint in ηn and j ≤ σi/ω include all edges c i a j .So at the moment G indeed satisfies 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 edge between c i and a j .One can easily check that G satisfies the degree sequence in the statement of the proposition.In particular, the vertices of degree 1 − ω+σ b n + σ ω k + ηn are c k , . . ., c k+rbηn .Define A := {a 1 , . . ., a σk/ω } and C := {c 1 , . . ., c k+rbηn }.Note that there are no edges between C and V 1 \ A in G. Claim 3.5.Let T be a B-tiling of G. Then T does not cover at least 3ηn/2 vertices in C.
Firstly, consider any copy B of B in T that contains at least one vertex in V r+1 .Since C is an independent set in G, observe that B contains at most ω vertices from C. Thus, there are at most ω|V r+1 | = ω(r − 1)ηn + ω(r − 2) vertices in C covered by copies of B in T that each contain at least 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 vertices from 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 1 \ A in G, B contains at least σ vertices in A.
Proof.Let G be the complete r-partite graph on n vertices with vertex classes V 1 , . . ., V r such that Then G satisfies the degree sequence condition in the proposition.The choice in size of V 1 ensures that any H-tiling in G covers at most |V 1 |h/σ < n − ηn vertices, as desired.

3.3.
A possible strengthening of Theorem 1.7.Whilst Proposition 3.3 demonstrates that we cannot lower every term in the degree sequence condition in Theorem 1.7 by much, perhaps one can cap the degrees as follows.
Question 3.7.Can the degree sequence condition in Theorem 1.7 be replaced by where C is a constant dependent only on H?
Note that Theorem 1.8 does not quite imply the Kühn-Osthus theorem (Theorem 1.4) due to the ηn-terms.On the other hand, an affirmative answer to Question 3.7, together with an analogous 'capped' version of Theorem 1.8 (ii), would fully imply the upper bound in Theorem 1.4.

Auxiliary results
4.1.The regularity and blow-up lemmas.The results in this section will be employed in our proof of Theorem 1.9.First we need the following definitions.The following groundbreaking result of Szemerédi [19] will be instrumental in our proof of Theorem 1.9.
Lemma 4.3 (Degree form of Szemerédi's Regularity lemma [19]).Let ε ∈ (0, 1) and M ∈ N. Then there exist natural numbers M and n 0 such that for any graph G on n ≥ n 0 vertices and any d ∈ (0, 1) there is a partition of the vertices of G into subsets V 0 , V 1 , . . ., V k and a spanning subgraph G of G such that the following hold: • For all 1 ≤ i, j ≤ k with i = j the pair (V i , V j ) G is ε-regular and has density either 0 or at least d.
We call V 1 , . . ., V k the clusters of our partition, V 0 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 set is V 1 , . . ., 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 |V (R)| = k.
We will apply the following well known fact, in conjunction with Lemma 4.5 (below), in Section 7.2.Lemma 4.5 (Key lemma [13]).Suppose that 0 < ε < d, that q, t ∈ N and that R is a graph with V (R) = {v 1 , . . ., v k }.We construct a graph G as follows: Replace every vertex v i ∈ V (R) with a set 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 0 := δ ∆ /(2 + ∆).If ε ≤ ε 0 and t − 1 ≤ ε 0 q then there are at least Let G be a graph as in Theorem 1.9 and R a reduced graph of G.The next well known lemma essentially says that R 'inherits' the degree sequence of G.
Let R be the reduced graph of G with parameters ε, M and d and set Let G and H be graphs and R be a reduced graph of G. Let H be a perfect H-tiling in R. The following result ensures that after removing only a few vertices from each cluster in R every edge in each copy of H ∈ H corresponds to a superregular pair.This will be essential to apply Lemma 4.8 in Section 7.4.Proposition 4.7 (See e.g.[16]).Let G be a graph, ε, d ∈ (0, 1) and M , ∆ ∈ N. Apply Lemma 4.3 to G with parameters ε, M and d to obtain a reduced graph R with clusters of size q.Let H be a subgraph of the reduced graph R with ∆(H) ≤ ∆ and label the vertices of The following fundamental result of Komlós, Sárközy and Szemerédi [11], known as the Blow-up lemma, essentially says that (ε, d)-superregular pairs behave like complete bipartite graphs with respect to containing bounded degree subgraphs.Lemma 4.8 (Blow-up lemma [11])).Given a graph F on vertices {1, . . ., f } and d, ∆ > 0, there exists an ε 0 = ε 0 (d, ∆, f ) > 0 such that the following holds.Given L 1 , . . ., L f ∈ N and ε ≤ ε 0 , let F * be the graph obtained from F by replacing each vertex i ∈ F with a set V i of L i new vertices and joining all vertices in V i to all vertices in V j whenever ij is an edge of F .Let G be a spanning subgraph 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) ≤ ∆. 4.2.Tilings in complete graphs.In [14], the following result of Kühn and Osthus is essential to their proof of Theorem 1.5.Lemma 4.9.[15, Lemma 12] Let H be a graph with χ(H) =: r ≥ 2 and hcf(H) = 1.Let h := |H| and ω(H) := (h − σ(H)) /(r − 1).Let 0 < β 1 λ 1 σ(H)/ω(H), 1 − σ(H)/ω(H), 1/h be positive constants.Suppose that F is a complete r-partite graph with vertex classes U 1 , . . ., U r such that: We will use the Blow-up lemma in tandem with the following generalisation of Lemma 4.91 to conclude 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 ≥ 2 and hcf(H) = 1.Let h := |H|.Set σ ∈ R such that σ(H) ≤ σ < h/r and ω := (h − σ) /(r − 1).Let 0 < β 2 λ 2 σ/ω, 1 − σ/ω, 1/h be positive constants.Suppose that F is a complete r-partite graph with vertex classes U 1 , . . ., U r such that: Proof.Note we may assume that σ > σ(H) as otherwise the result follows immediately from Lemma 4.9.We choose β 2 β 1 λ 2 λ 1 where β 1 and λ 1 are as in Lemma 4.9.Additionally we may assume β 2 , λ 2 (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 graph on rh vertices (that is, each vertex class of H * has size h).Observe that H * has a perfect H-tiling using precisely r copies of H.
Repeatedly delete disjoint copies of H * from F (and therefore update the classes U 1 , . . ., U r ) until the first point for which there is some i < r such that (1 − λ Call the resulting graph F .Note that σ/ω > σ(H)/ω(H), so we can indeed obtain F .Further note that our (sufficiently small) choice of β 2 ensures each class U j still contains at least a β 1/2 2 -proportion of the vertices it started with.So now for all j < r.Thus, by Lemma 4.9, F contains a perfect H-tiling and therefore, so too does F , as desired.

4.3.
A degree sequence Komlós theorem.In [14], Kühn and Osthus begin their proof of Theorem 1.5 by applying Komlós' theorem (Theorem 1.3).In our proof of Theorem 1.9 we will use the following degree sequence version of Komlós' theorem that the authors and Liu proved in [7].Set σ ∈ R such that σ(H) ≤ σ ≤ h/r and ω := (h − σ) /(r − 1).Then there exists an integer n 0 = n 0 (η, σ, H) ∈ N such that the following holds.Suppose G is a graph on n ≥ n 0 vertices with degree sequence d 1 ≤ . . .≤ d n such that Then G contains an H-tiling covering all but at most ηn vertices.

4.4.
Bézout's Lemma.To prove Theorem 5.2 we will need the following elementary arithmetic result.

5.
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 7.3.2 of the proof of Theorem 1.9.At the beginning of Section 7.3, we will have a certain B-tiling B of a reduced graph R (the graph B will be defined later).Denote the copies of B in B by B1 , B2 , . . ., Bk .For applications of Lemma 4.10 required at the end of our proof of Theorem 1.9, we will need |V G ( Bi )| to be divisible by h for each 1 ≤ i ≤ k.The following theorem is the crucial tool 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 that given an optimal colouring c ∈ C H we let x c,1 ≤ x c,2 ≤ . . .≤ x c,r denote the sizes of the colour classes of c.We require the following definitions.
• if r ≥ 3 and hcf χ (H) = 1, there exists a collection of non-negative integers {a p,c,i : and For each 1 ≤ p ≤ r − 1, c ∈ C H and j ∈ {1, . . ., r}, let Z p,c,j be the multiset defined by the following table : Colour class size The following fact will be useful in our proof of Theorem 5.2.
Fact 5.3.For any 1 ≤ 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 1 , t 2 , . . ., t b .Since hcf c (H) = 1, by Bezout's Lemma (Lemma 4.12) there exist integers a 1 , . . ., a b such that b i=1 For each 1 ≤ i ≤ b, take a i := a i + â and ā := 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 Observe that constructing such multisets X c immediately yields a collection of non-negative integers {a p,c,i : c ∈ C H , 1 ≤ 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.
In order to start constructing our multisets X c , we define the following multiset: 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 − 1 integers b c,j such that the following holds: (1) We now construct our multisets X c .Fix c ∈ C H . Choose t c ∈ N to be the smallest natural number such that Let Y c be the multiset defined by the following table : Colour class size for each 1 ≤ i ≤ r. 4) 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 a partition of Z p,c,J into {x c,L } and r p-subsets contained in D c .5) Place into Q these r p-subsets contained in D c .6) 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.In particular, it is easy to see that Y c now has a partition Q Yc into p-subsets contained in D c .Let t c be the number of collections of p copies of H added during the algorithm and tc := t c + t c .Then the multiset Ŷc , defined by the table below, can be partitioned into p-subsets contained in D c using the partition Take X c := Ŷc .We now confirm that our multisets X c satisfy By (1) and the definition of X c for each c ∈ C H we have ≡ 1 mod h.
Therefore, recalling the discussion earlier in this proof, there must exist the desired collection of non-negative integers {a p,c,i : c ∈ C H , 1 ≤ i ≤ z p }, and we take ā to be the maximum element in this collection.

Proof Overview
The rest of this paper will be devoted to proving Theorem 1.9 and here we outline the proof.As noted in Section 1.2, our proof follows closely Kühn 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 − 1).Note that it suffices to prove the result in the case when σ ∈ Q.First we define a bottle graph B that contains a perfect H-tiling.
. Define B to be the r-partite bottle graph with neck σĉ and width ωĉ (note that σĉ, ωĉ ∈ N).Observe that |B| = hĉ; σ(B) = σĉ; ω(B) = ωĉ.Further, Since |B| = hĉ; σ(B) = σĉ; ω(B) = ωĉ, we have that G satisfies the hypothesis of the degree sequence Komlós 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 ĉ implies that B has a perfect H-tiling consisting of ĉ copies of H. Indeed, this follows as B has a perfect tiling of a − σ(H)b copies of K h,...,h and hb − ar copies of H, where K h,...,h is the complete r-partite graph with each vertex class having size h.Ideally one would like to use B as a framework to build the perfect H-tiling in G.However, as explained 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 where σ(1+λ)s/ω ∈ N. Recall that σ < ω.Let B be the r-partite bottle graph with neck σ(1 + λ)s/ω and width s. 3  Moreover, we choose λ and s such that B contains a perfect B-tiling.Hence B contains a perfect H-tiling.Note that Since λ is chosen to be small (and so χ cr ( B) is very close to χ cr (B)), one can still apply Theorem 4.11 on inputs B and R.That is, R contains an almost perfect B-tiling B. Denote the copies of B in B by B1 , B2 , . . ., Bk .By removing a small number of vertices from each cluster in R, we can ensure the edges of each Bi correspond to superregular pairs.Let V 0 denote the set of all vertices in G not contained in the clusters lying in the tiling B.
For each 1 ≤ i ≤ k, let Ĝi be the r-partite subgraph of G whose jth vertex class is the union of all those clusters contained in the jth vertex class of Bi , for each 1 ≤ j ≤ r.Let G * i be the complete r-partite graph on the same vertex set as Ĝi .We introduce the graph B (rather than just working with B) since B has the following crucial property: For each 1 ≤ i ≤ k we can arbitrarily delete a small number of vertices from G * i (and correspondingly Ĝi ) and, provided |V (G * i )| is now divisible by h, the resulting graph satisfies the hypothesis of Lemma 4.10.That is, this graph contains a perfect H-tiling.Then the Blow-up lemma (Lemma 4.8) implies that each Ĝi contains a perfect H-tiling.
We make use of this property of B as follows: In Section 7.2 we remove copies of H from G that cover all vertices in V 0 , as well as a small (possibly zero) number of vertices from each Ĝi ; call this H-tiling (formed from these copies of H) H 1 .Deleting these covered vertices from each Ĝi , if ) is still divisible by h for each 1 ≤ i ≤ k then each Ĝi now contains a perfect H-tiling (by our argument above).However, for some i, we may have that |V ( Ĝi )| is not divisible by h.So in Section 7.3 we remove a further bounded number of copies of H, forming an H-tiling ) is divisible by h for each 1 ≤ i ≤ k.Thus, we now have that each Ĝi contains a perfect H-tiling Ĥi .Combining the tilings H 1 , H 2 , Ĥ1 , . . ., Ĥk 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) ≥ 3 and the latter when H is bipartite.This is where our proof differs the most from that in [14] as we must make use of Theorem 5.2 to find suitable copies of H to ensure each |V ( Ĝi )| is divisible by h.

7.
Proof of Theorem 1.9 7.1.Applying the regularity lemma.Note that it suffices to prove the theorem in the case when σ ∈ Q.Let n be sufficiently large and fix constants that satisfy the following hierarchy As discussed in Section 6, we choose s ∈ N sufficiently large and define B to be the r-partite bottle graph with neck σ(1 + λ)s/ω and width s.As before, we choose λ and s such that B contains a perfect B-tiling, which implies that B contains a perfect H-tiling.Note again that χ cr ( B) = r − 1 + σ(1 + λ)/ω. 3We have that σ(1 + λ)/ω < 1 by our choice of λ and that σ < ω.
Moreover, choose η 1 and M such that where n 0 is defined as in Theorem 4.11.Let G be an n-vertex graph as in the statement of Theorem 1.9.Apply Lemma 4.3 with parameters ε, d and M to G to obtain clusters V 1 , . . ., V k , an exceptional set V 0 and a pure graph G , where q := |V 1 | = . . .= |V k | and k ≥ M .Let R be the corresponding reduced graph.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 For 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 ) and ( 4) holds, we apply Theorem 4.11 to find a Btiling B covering all but at most η 1 k vertices in R. Denote the copies of B in B by B1 , B2 , . . ., Bk .Now delete all clusters not contained in some Bi from R and add the vertices in these clusters to V 0 .Therefore now From now on, we denote by R the subgraph of the reduced graph induced by all the remaining clusters and redefine k : For each Bi in B, let B i be a perfect B-tiling in Bi (recall that earlier we chose s and λ such that B contains a perfect B-tiling).Let B := B i and observe that B is a perfect B-tiling in R. To aid with calculations we will sometimes work with B instead of B.
Let q := (1−ε| B|)q.By Proposition 4.7, for all 1 ≤ i ≤ k we can remove ε| B|q vertices from each cluster V a belonging to Bi so that each edge V a V b in Bi now corresponds to a (2ε, d/2)-superregular pair (V a , V b ) G .Further, all clusters now have size q and for each edge V a V b in Bi the pair (V a , V b ) G is a 2ε-regular pair with density at least d/2 (by Fact 4.4).Add all the vertices we removed from the clusters to V 0 and observe that now, since ε 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 can obtain the following partition of each cluster.Apply Claim 7.1 to every cluster to yield a partition of V (G) − V 0 into red and blue vertices.In the next section, we will remove vertices of particular copies of H in G from their respective clusters and do so in such a way that we avoid all the red vertices of G.After removing these vertices, we will be able to conclude that each (modified) pair (V a , V b ) G is (5ε, d/5)-superregular4 since V red a and V red b had no vertices removed from them.After the next section, we will only remove a bounded number of vertices from the clusters, which will not affect the superregularity of pairs of clusters in any significant way.7.2.Covering the exceptional vertices.As in [14], given x ∈ V 0 , we call a copy of B ∈ B useful for x if there exist r − 1 clusters in 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 are useful for x.The following calculation demonstrates that By ( 2) and ( 6), we have that Rearranging, we get Since α η, we have that

Now as |B|q ≥ n
2|B| and η 1 β, η, 1/h we have that Hence we can assign greedily each vertex x ∈ V 0 to a copy B x that is useful for x and do so in such a way that at most βq vertices in V 0 are assigned to the same copy B ∈ B. Then for each copy B x ∈ B that is useful for some x ∈ V 0 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 − 1 clusters belonging to different vertex classes of B x , Claim 7.1 implies that x has at least αq /4 blue neighbours in each of these r − 1 clusters.Further, we can find αq /4 blue vertices in a cluster belonging to the vertex class of B x that does not necessarily contain any neighbours of x.Then it is easy to see that we can find subclusters S 1 , . . ., S r of r clusters in B x such that: all vertices in S 1 ∪ . . .∪ S r are blue vertices; |S i | = αq /4 for each i; every vertex in S 1 ∪ . . .∪ S r−1 is a neighbour of x in G.By Fact 4.4, each pair (S i , S j ), 1 ≤ i < j ≤ r, corresponds to an (8ε/α)-regular pair in G with density at least d/3.Using Lemma 4.5 with parameters 8ε/α, d/3, αq /4, h − 1, we find a copy of H containing x. Since each B ∈ B has been assigned to at most βq vertices in V 0 and β α (from (2)), we may repeat the above argument to find copies of H that contain each exceptional vertex in such a way that the copies are disjoint and contain no red vertices.Denote the H-tiling induced by these copies of H by H 1 .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.

7.3.
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 Bi in B to find an H-tiling that covers every vertex of V G ( Bi ).Combining these H-tilings with H 1 will result in a perfect H-tiling in G as desired.Recall that, for each 1 ≤ i ≤ k, Ĝi is the r-partite subgraph of G whose jth vertex class is the union of all those clusters contained in the jth vertex class of Bi , for each 1 ≤ j ≤ r.Further, recall that G * i is the complete r-partite graph on the same vertex set as Ĝi .To apply Lemma 4.8 to each Bi in B we require that each G * i contains a perfect H-tiling.To guarantee the existence of these perfect H-tilings we will apply Lemma 4.10.To use Lemma 4.10 on G * i we require that |V ( Ĝ * i )| is divisible by h.When we first chose our B-tiling this was the case.Indeed, as each Bi contained a perfect H-tiling and every cluster V i was the same size, |V (G * i )| was divisible by h.However, in the last section we took out vertices from G in a greedy way, changing the sizes of the clusters in R. Hence we cannot guarantee that |V (G * i )| is still divisible by h for each i.Now we will take out a further bounded number of copies of We now split into two cases: when r ≥ 3 and when r = 2.When r ≥ 3 we have that hcf χ (H) = 1 and this property will be central to our argument.For r = 2, we have that hcf c (H) = 1 and hcf 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 the property that hcf χ (H) ≤ 2 in our argument.The only time we (implicitly) use the property that hcf χ (H) ≤ 2 will be when we apply Lemma 4.10.7.3.1.Case 1: r ≥ 3.For a subgraph S of R, let V R (S) denote the vertex set of S. To assist in our argument, we define an auxiliary graph F whose vertices are the copies of B in B and for B 1 , B 2 ∈ V (F ), we let B 1 B 2 be an edge in F if and only if there exists a vertex x in V R (B 1 ) and r − 1 vertices in V R (B 2 ), all in different vertex classes of B 2 , (or vice versa) such that these r vertices induce a K r in R. Assume F is connected and let B 1 B 2 be an edge in F .Then we may apply Lemma 4.5 to find h − 1 disjoint copies of H which each have one vertex in V G (B 1 ) and all other vertices in V G (B 2 ) (or vice versa).This means that we can remove at most h − 1 copies of H to ensure V G (B 1 ) is divisble by h.Continuing in this way we can 'shift the remainders mod h' along a spanning tree of 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 vertices in R belonging to copies of B in C and V G (C) for the union of the clusters corresponding to the vertices in V R (C).In what follows our aim is to remove a bounded number of copies of H to ensure that for each component C ∈ C we have that |V G (C)| is divisible by h.Then we can apply our previous argument 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/4)k big.If a vertex is not big, call it small.Note by ( 5) that all but at most ωk/h − 1 vertices in R are big.
Proof.Recall that B has width ωĉ.Suppose Claim 7.2 is false.Then there exists some Thus a must have neighbours in at least r − 1 vertex classes of B 0 .We can therefore construct a copy of K r in R which consists of a together with r − 1 of its neighbours in B 0 .But by definition of the auxiliary graph F , we must have that B 0 is adjacent in F to the copy of B in B that contains a.This contradicts that C 1 and C 2 were different components of F .Thus Claim 7.2 holds.
Proof.Take some big vertex )k, as otherwise x has a neighbour in R outside of C x and we are done.Since r ≥ 3, Since there are at most ωk/h − 1 small vertices in R, we have that z has a neighbour w which is a big vertex.But then w ∈ V R (C x ).Thus Claim 7.3 holds.Claim 7.4.There exists a copy K of K r in R which has vertices in at least two components of F .
If r = 3, then we choose x 3 in the common neighbourhood of x 1 and x 2 , and we are done.So assume r ≥ 4. Since there are at most ωk/h small vertices, we can choose a big vertex x 3 in the common neighbourhood of x 1 and x 2 .Then x 1 , x 2 and x 3 have a common neighbourhood of size at least ((r − 4)ω/h + 3η/4)k.
If r = 4, then we choose x 4 in the common neighbourhood of x 1 , x 2 and x 3 and we are done.Otherwise r ≥ 5 and we continue as before.Thus Claim 7.4 holds.
For such a copy K of K r in R, we now show that we can take out a bounded number of copies of H from the clusters corresponding to the vertices of K in such a way that that leaves one of the components C ∈ C with |V G (C)| divisible by h.We use Theorem 5.2 and Lemma 4.5 to achieve this.We will then repeat this process to ensure |V G (B)| is divisible by h for each B ∈ B. Denote this H-tiling (formed from these copies of H) by H 2 .
We now have all we need to apply Lemma 4.8 to find a perfect H-tiling Ĥi in Ĝi for each 1 ≤ i ≤ k.Then H 1 ∪ H 2 ∪ Ĥ1 ∪ . . .∪ Ĥk is a perfect H-tiling in G. Hence we have proved Theorem 1.9.

Acknowledgements
The first author would like to thank Pat Devlin for a helpful conversation at Building Bridges II.The authors are also grateful to the referees for their helpful and careful reviews.

Definition 4 . 1 .Definition 4 . 2 .
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| .Set 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.Given ε > 0, 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.

Definition 5 . 1 .
Let H be a graph with χ(H) =: r.Fix 1 ≤ p ≤ r − 1.For each c ∈ C H , define D c to be the multiset [x c,1 , x c,2 , . . ., 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 1 , x c,j 2 , . . ., x c,jp ] where j 1 , j 2 , . . ., j p ∈ {1, . . ., r} are distinct.Let z p := r p be the number of p-subsets contained in D c .For each colouring c ∈ C H , label the p-subsets contained in D c by A p,c,1 , A p,c,2 , . . ., A p,c,zp .Let S p,c,J := x∈A p,c,J x for each c ∈ C H , 1 ≤ J ≤ z p .Theorem 5.2.Let H be an r-partite graph and let h := |H|.Fix 1 ≤ p ≤ r − 1.Let b be the number of components of H and t 1 , . . ., 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 ā ∈ N such that a i ≤ ā for all 1 ≤ i ≤ b, and b i=1

r− 1 1 ThenDefinition 5 . 4 .
pt c + b c,r−|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,1 , . . ., x c,r in Y c must be sufficiently different from one another.We employ the following algorithm which transforms Y c into a multiset which can be partitioned into p-subsets contained in D c using Fact 5.3.To state the algorithm we require the following definition.For each c

Claim 7 . 1 .
Let V a be a cluster.Then there exists a partition of V a into a red part V red a and a blue part V blue a such that ||V red a | − |V blue a || ≤ εq and ||N G (x) ∩ V red a | − |N G (x) ∩ V blue a || < εq for all x ∈ V (G).
contains any big vertex y, then y has a neighbour inV R (C x ) since |R \ V R (C x )| < (1 − ω/h + η/4)k and we are done.Hence assume all big vertices are in V R (C x ).Then all vertices in R \ V R (C x ) are small vertices.Let z be a small vertex in R \ V R (C x ).Since r ≥ 3, d R (z) ≥ (1 − (ω + σ)/h + η/4)k ≥ (ω/h + η/4)k.

Claim 7 . 5 .+ σ h k 1 + σ ω i + ηk 1 4 for all 1 ≤
There exists t ∈ N such that by removing at most t + (|B| − |C|)(h − 1) 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)| is divisible by h.Apply Claim 7.4 to find a copy K of K r in R which has vertices in at least two components of F .Let C * be a component of F which contains at least one vertex of K .Let p be the number of vertices of K contained in C * and observe that 1≤ p ≤ r − 1.Let 0 ≤ g ≤ h − 1 such that |V G (C * )| ≡ g mod h.If g = 0 then |V G (C * )|is divisible by h and we consider the graphsF 1 := F − V (C * ) and R 1 := R − V R (C * ).So assume 1 ≤ g ≤ h − 1.Observe that we can apply 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, given any colouring c of H and p-subset P contained in D c (recall Definition 5.1) we can find any bounded number of disjoint 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 .So there exists j ∈ {1, . . ., 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 would result in removing precisely S p,c,j vertices from V G (C * ).By Theorem 5.2, there exist a collection of non-negative integers {a p,c,i :c ∈ C H , 1 ≤ i ≤ z p } and ā ∈ N such that a p,c,i ≤ ā for all c ∈ C H , 1 ≤ i ≤ z p , c,i S p,c,i ≡ g mod h.Hence we can removeg • c∈C H zp i=1 a p,c,i ≤ (h − 1)ā|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 graphsF 1 := F − V (C * ) and R 1 := R − V R (C * ).Let k 1 := |R 1 |.Claim 7.2 and (5) together give us that R 1 has degree sequence d R 1 ,1 ≤ . . .≤ d R 1 ,k 1 where d R 1 ,i ≥ 1 − ω i ≤ ωk 1 h .Suppose |C| ≥ 3. Arguing as in Claims 7.3 and 7.4 we can find a copy K 1 of K r in R 1 which has vertices in at least two components of F 1 .Let C * * be a component of F which contains at least one vertex of K 1 .As before by removing at most (h − 1)ā|C H |z p copies of H from the clusters of K 1 we can make |V G (C * )| divisible by h.Since |G| is divisible by h, we can continue in this way to make |V G (C)| divisible by h for each component C ∈ C. We then apply the 'shifting the remainders 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| − 1)(h − 1)ā|C H |z p disjoint copies of H from G. Each time we use the 'shifting the remainders mod h' argument on a connected component C ∈ C we remove at most (|C| − 1)(h − 1) disjoint copies of H in G. Hence overall we remove at most (|C| − 1)(h − 1)ā|C H |z p + (|B| − |C|)(h − 1) disjoint copies of H in G.