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arXiv:1201.1882v1 [math.CO] 9 Jan 2012

A MULTIPARTITE HAJNAL-SZEMER´EDI THEOREM

PETER KEEVASH AND RICHARD MYCROFT

Abstract. The celebrated Hajnal-Szemer´ edi theorem gives the precise minimum degree

threshold that forces a graph to contain a perfect Kk-packing. Fischer’s conjecture states

that the analogous result holds for all multipartite graphs except for those formed by a

single construction. Recently, we deduced an approximate version of this conjecture from

new results on perfect matchings in hypergraphs. In this paper, we apply a stability analysis

to the extremal cases of this argument, thus showing that the exact conjecture holds for

any sufficiently large graph.

1. Introduction

A fundamental result of Extremal Graph Theory is the Hajnal-Szemer´ edi theorem, which

states that if k divides n then any graph G on n vertices with minimum degree δ(G) ≥

(k − 1)n/k contains a perfect Kk-packing, i.e. a spanning collection of vertex-disjoint k-

cliques. This paper considers a conjecture of Fischer [2] on a multipartite analogue of this

theorem. Suppose V1,...,Vkare disjoint sets of n vertices each, and G is a k-partite graph

on vertex classes V1,...,Vk(that is, G is a graph on the vertex set V1∪ ··· ∪ Vksuch that

no edge of G has both endvertices in the same Vj). We define the partite minimum degree

of G, denoted δ∗(G), to be the largest m such that every vertex has at least m neighbours

in each part other than its own, i.e.

δ∗(G) := min

i∈[k]min

v∈Vi

min

j∈[k]\{i}|N(v) ∩ Vj|,

where N(v) denotes the neighbourhood of v. Fischer conjectured that if δ∗(G) ≥ (k−1)n/k

then G has a perfect Kk-packing. This conjecture is straightforward for k = 2, as it is not

hard to see that any maximal matching must be perfect. However, Magyar and Martin [8]

constructed a counterexample for k = 3, and furthermore showed that their construction

gives the only counterexample for large n. More precisely, they showed that if n is sufficiently

large, G is a 3-partite graph with vertex classes each of size n and δ∗(G) ≥ 2n/3, then either G

contains a perfect K3-packing, or n is odd and divisible by 3, and G is isomorphic to the

graph Γn,3,3defined in Construction 1.2.

The implicit conjecture behind this result (stated explicitly by K¨ uhn and Osthus [6]) is

that the only counterexamples to Fischer’s original conjecture are the constructions given by

the graphs Γn,k,kdefined in Construction 1.2 when n is odd and divisible by k. We refer to

this as the modified Fischer conjecture. If k is even then n cannot be both odd and divisible

by k, so the modified Fischer conjecture is the same as the original conjecture in this case.

Martin and Szemer´ edi [9] proved that (the modified) Fischer’s conjecture holds for k = 4.

Another partial result was obtained by Csaba and Mydlarz [1], who gave a function f(k)

with f(k) → 0 as k → ∞ such that the conjecture holds for large n if one strengthens the

degree assumption to δ∗(G) ≥ (k − 1)n/k + f(k)n. Recently, an approximate version of the

conjecture assuming the degree condition δ∗(G) ≥ (k−1)n/k+o(n) was proved independently

and simultaneously by Keevash and Mycroft [5], and by Lo and Markstr¨ om [7]. The proof

in [5] was a quick application of the geometric theory of hypergraph matchings developed in

Date: January 10, 2012.

Research supported in part by ERC grant 239696 and EPSRC grant EP/G056730/1.

1

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2 PETER KEEVASH AND RICHARD MYCROFT

the same paper; this will be formally introduced in the next section. By a careful analysis

of the extremal cases of this result, we will obtain the following theorem, the case r = k of

which shows that (the modified) Fischer’s conjecture holds for any sufficiently large graph.

Note that the graph Γn,r,kin the statement is defined in Construction 1.2.

Theorem 1.1. For any r ≥ k there exists n0 such that for any n ≥ n0 with k | rn the

following statement holds. Let G be a r-partite graph whose vertex classes each have size n

such that δ∗(G) ≥ (k − 1)n/k. Then G contains a perfect Kk-packing, unless rn/k is odd,

k | n, and G∼= Γn,r,k.

We now give the generalised version of the construction of Magyar and Martin [8] showing

Fischer’s original conjecture to be false.

Construction 1.2. Suppose rn/k is odd and k divides n. Let V be a vertex set partitioned

into parts V1,...,Vr of size n. Partition each Vi, i ∈ [r] into subparts Vj

n/k. Define a graph Γn,r,k, where for each i,i′∈ [r] with i ?= i′and j ∈ [k], if j ≥ 3 then

any vertex in Vj

then any vertex in Vj

i, j ∈ [k] of size

iis adjacent to all vertices in Vj′

iis adjacent to all vertices in Vj′

i′ with j′∈ [k] \ {j}, and if j = 1 or j = 2

i′ with j′∈ [k] \ {3 − j}.

V1

V2

V3

?

V1

?

V2

?

V3

Figure 1. Construction 1.2 for the case k = r = 3.

Figure 1 shows Construction 1.2 for the case k = r = 3. To avoid complicating the

diagram, edges between V1and V3are not shown: these are analogous to those between V1

and V2and between V2and V3. For n = k this is the exact graph of the construction; for

larger n we ‘blow up’ the graph above, replacing each vertex by a set of size n/k, and each

edge by a complete bipartite graph between the corresponding sets. In general, it is helpful

to picture the construction as an r by k grid, with columns corresponding to parts Vi, i ∈ [r]

and rows Vj=?

vertices have neighbours in other columns in their own row and other rows besides rows V1

and V2. Thus δ∗(G) = (k − 1)n/k. We claim that there is no perfect Kk-packing. For any

Kkhas at most one vertex in any Vjwith j ≥ 3, so at most k −2 vertices in?

Kkmust have k − 2 vertices in?

i∈[r]Vj

i, j ∈ [k] corresponding to subparts of the same superscript. Vertices

have neighbours in other rows and columns to their own, except in rows V1and V2, where

j≥3Vj. Also

????

j≥3Vj??? = (k − 2)rn/k, and there are rn/k copies of Kkin a perfect packing. Thus each

j≥3Vj, and so 2 vertices in V1∪ V2, which must either

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A MULTIPARTITE HAJNAL-SZEMER´EDI THEOREM3

both lie in V1or both lie in V2. However, |V1| = rn/k is odd, so V1cannot be perfectly

covered by pairs. Thus G contains no perfect Kk-packing.

This paper is organised as follows. In the next section we introduce ideas and results

from [5] on perfect matchings in k-graphs. Section 3 gives an outline of the proof of The-

orem 1.1. In Sections 4 to 7 we prove several preliminary lemmas, before combining these

lemmas in Section 8 to prove Theorem 1.1.

Notation. The following notation is used throughout the paper: [k] = {1,...,k}; if X

is a set then

?X

statements with more variables are defined similarly). If x is a vertex in a graph then N(x)

is the neighbourhood of x.

k

?

is the set of subsets of X of size k; x ≪ y means that for every y > 0

there exists some x0> 0 such that the subsequent statement holds for any x < x0(such

2. Perfect matchings in hypergraphs

In this section we describe the parts of the geometric theory of perfect matchings in

hypergraphs from [5] that we will use in the proof of Theorem 1.1. We start with some

definitions. A hypergraph G consists of a vertex set V and an edge set E, where each edge

e ∈ E is a subset of V . We say that G is a k-graph if every edge has size k. A matching M in

G is a set of vertex-disjoint edges in G. We call M perfect if it covers all of V . We identify

a hypergraph H with its edge set, writing e ∈ H for e ∈ E(H), and |H| for |E(H)|. A

k-system is a hypergraph J in which every edge of J has size at most k and ∅ ∈ J. We refer

to the edges of size r in J as r-edges of J, and write Jrto denote the r-graph on V (J) formed

by these edges. A k-complex J is a k-system whose edge set is closed under inclusion, i.e. if

e ∈ J and e′⊆ e then e′∈ J. For any non-empty k-graph G, we may generate a k-complex

G≤whose edges are any e ⊆ V (G) such that e ⊆ e′for some edge e′∈ G.

Let V be a set of vertices, and let P partition V into parts V1,...,Vrof size n. Then we

say that a hypergraph G with vertex set V is P-partite if |e ∩ Vi| ≤ 1 for every i ∈ [r] and

e ∈ G. We say that G is r-partite if it is P-partite for some partition P of V into r parts.

Let J be a P-partite k-system on V . For each 0 ≤ j ≤ k − 1 we define the partite

minimum j-degree δ∗

j(J) as the largest m such that any j-edge e has at least m extensions

to a (j + 1)-edge in any part not intersected by e, i.e.

δ∗

j(J) := min

e∈Jj

min

i:e∩Vi=∅|{v ∈ Vi: e ∪ {v} ∈ J}|.

The partite degree sequence is δ∗(J) = (δ∗

dependence on P in our notation: this will be clear from the context. Note also that this

is not the standard notion of degree used in k-graphs, in which the degree of a set is the

number of edges containing it. Our minimum degree assumptions will always be of the form

δ(J) ≥ a pointwise for some vector a = (a0,...,ak−1), i.e. δi(J) ≥ aifor 0 ≤ i ≤ k − 1.

It is helpful to interpret this ‘dynamically’ as follows: when constructing an edge of Jkby

greedily choosing one vertex at a time, there are at least aichoices for the (i + 1)st vertex

(this is the reason for the requirement that ∅ ∈ J, which we need for the first choice in the

process).

The following key definition relates our theorems on hypergraphs to graphs. Fix r ≥ k

and a partition P of a vertex set V into r parts V1,...,Vrof size n. Let G be an P-partite

graph on V . Then the clique k-complex J(G) of G is the k-complex whose edges of size

i are precisely the copies of Ki in G for 0 ≤ i ≤ k. Note that J(G) must be P-partite.

0(J),...,δ∗

k−1(J)). Note that we suppress the

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4 PETER KEEVASH AND RICHARD MYCROFT

Furthermore, if δ∗(G) ≥ (k − 1)n/k − αn and 0 ≤ i ≤ k − 1, then the vertices of any copy

of Kiin G have at least n − in/k − iαn common neighbours in each Vjwhich they do not

intersect. That is, if G satisfies δ∗(G) ≥ (k − 1)n/k − αn, then the clique k-complex J(G)

satisfies

?

Note also that any perfect matching in the k-graph J(G)k corresponds to a perfect Kk-

packing in G. So if we could prove that any P-partite k-complex J on V which satisfies (1)

must have a perfect matching in the k-graph Jk, then we would have already proved Theo-

rem 1.1! Along these lines, Theorem 2.4 in [5] shows that any such J must have a match-

ing in Jk which covers all but a small proportion of the vertices of J. (Here we assume

1/n ≪ α ≪ 1/r,1/k). However, two different families of constructions show that this con-

dition does not guarantee a perfect matching in Jk; we refer to these as space barriers and

divisibility barriers. We will describe these families in some detail, since the results of [5]

show that these are essentially the only k-complexes J on V which satisfy (1) but do not

have a perfect matching in Jk. Firstly, space barriers are characterised by a bound on the

size of the intersection of every edge with some fixed set S ⊆ V (J). If S is too large, then Jk

cannot contain a perfect matching. The following construction gives the precise formulation.

(1)δ∗(J(G)) ≥

n,

?k − 1

k

− α

?

n,

?k − 2

k

− 2α

?

n,...,

?1

k− (k − 1)α

?

n

?

.

Construction 2.1. (Space barriers) Suppose P partitions a set V into r parts V1,...,Vrof

size n. Fix j ∈ [k−1] and a set S ⊆ V containing s = ⌊(j/k + α)n⌋ vertices in each part Vj.

Then we denote by J = Jr(S,j) the k-complex in which Ji(for 0 ≤ i ≤ k) consists of all

P-partite sets e ⊆ V of size i that contain at most j vertices of S. Observe that δ∗

for 0 ≤ i ≤ j − 1 and δ∗

matching in Jkhas size at most

?|V \S|

Having described the general form of space barriers, we now turn our attention to di-

visibility barriers. These are characterised by every edge satisfying an arithmetic condition

with respect to some partition Q of V . To be more precise, we need the following definition.

Fix any partition Q of a vertex set V into d parts V1,...,Vd. For any Q-partite set S ⊆ V

(that is, S has at most one vertex in each part of Q), the index set of S with respect to Q

is iQ(S) := {i ∈ [d] : |S ∩ Vi| = 1}. For general sets S ⊆ V , we have the similar notion of

the index vector of S with respect to Q; this is the vector iQ(S) := (|S ∩ V1|,...,|S ∩ Vd|)

in Zd. So iQ(S) records how many vertices of S are in each part of Q. Observe that if S is

Q-partite then i(S) is the characteristic vector of the index set i(S). When Q is clear from

the context, we write simply i(S) and i(S) for iQ(S) and iQ(S) respectively, and refer to

i(S) simply as the index of S. We will consider the partition Q to define the order of its

parts so that iQ(S) is well-defined.

i(J) = n

i(J) = n − s for j ≤ i ≤ k − 1, so (1) is satisfied. However, any

k−j

and so leaves at least r(αn−k) vertices uncovered.

?

Construction 2.2. (Divisibility barriers) Suppose Q partitions a set V into d parts, and L

is a lattice in Zd(i.e. an additive subgroup) with i(V ) / ∈ L. Fix any k ≥ 2, and let G be the

k-graph on V whose edges are all k-tuples e with i(e) ∈ L. For any matching M in G with

vertex set S =?

For the simplest example of a divisibility barrier take d = 2 and L = ?(−2,2),(0,1)?. So

(x,y) ∈ L precisely when x is even. Then the construction described has |V1| odd, and the

e∈Me we have i(S) =?

e∈Mi(e) ∈ L. Since we assumed that i(V ) / ∈ L it

follows that G does not have a perfect matching.

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A MULTIPARTITE HAJNAL-SZEMER´EDI THEOREM5

edges of G are all k-tuples e ⊆ V such that |e ∩ V1| is even. If |V | = n and |V1| ∼ n/2,

then any set of k −1 vertices of G is contained in around n/2 edges of G, but G contains no

perfect matching.

We now consider the multipartite setting. Let P partition a vertex set V into parts

V1,...,Vrof size n, and let Q be a partition of V into d parts U1,...,Udwhich refines P.

Then we say that a lattice L ⊆ Zdis complete with respect to P if L contains every difference

of basis vectors ui−ujfor which Uiand Ujare contained in the same part Vℓof P, otherwise

we say that L is incomplete with respect to P. The idea behind this definition is that if L

is incomplete with respect to P, then it is possible that iQ(V ) / ∈ L, in which case we would

have a divisibility barrier to a perfect matching, whilst if L is complete with respect to P

then this is not possible. There is a natural notion of minimality for an incomplete lattice

L with respect to P: we say that Q is minimal if L does not contain any difference of basis

vectors ui−ujfor which Ui,Ujare contained in the same part Vℓof P. For suppose L does

contains some such difference ui− ujand form a partition Q′from Q by merging parts Ui

and Uj of Q. Let L′⊆ Zd−1be the lattice formed by this merging (that is, by replacing

the ith and jth coordinates with a single coordinate equal to their sum). Then L′is also

incomplete with respect to P, so we have a smaller divisibility barrier.

Let J be an r-partite k-complex whose vertex classes V1,...,Vreach have size n. The next

theorem, Theorem 2.9 from [5], states that if J satisfies (1) and Jkis not ‘close’ to either a

space barrier or a divisibility barrier, then Jkcontains a perfect matching. Moreover, we can

find a perfect matching in Jkwhich has roughly the same number of edges of each index.

More precisely, for a perfect matching M in Jkand a set A ∈?[r]

all A ∈?[r]

notion of closeness to a space or divisibility barrier as follows. Let G and H be k-graphs on

a common vertex set V of size n. We say G is β-contained in H if all but at most βnkedges

of G are edges of H. Also, given a partition P of V into d parts, we define the µ-robust edge

lattice Lµ

least µnkedges e ∈ G with iP(e) = v.

Theorem 2.3. Suppose that 1/n ≪ γ ≪ α ≪ µ,β ≪ 1/r, r ≥ k and k | rn. Let P′partition

a set V into parts V1,...,Vreach of size n. Suppose that J is a P′-partite k-complex with

δ∗(J) ≥

k

k

?let NA(M) be the number

of edges e ∈ M with index i(e) = A. We say that M is balanced if NA(M) is constant over

k

?, that is, if there are equally many edges of each index. Similarly, we say that

M is γ-balanced if NA(M) = (1 ± γ)NB(M) for any A,B ∈?[r]

k

?. Finally, we formalise the

P(G) ⊆ Zdto be the lattice generated by all vectors v ∈ Zdsuch that there are at

?

n,

?k − 1

− α

?

n,

?k − 2

k

− α

?

n,...,

?1

k− α

?

n

?

.

Then J has at least one of the following properties.

1 (Matching): Jkcontains a γ-balanced perfect matching.

2 (Space barrier): Jkis β-contained in Jr(S,j)k for some j ∈ [k − 1] and S ⊆ V

with ⌊jn/k⌋ vertices in each Vi, i ∈ [r].

3 (Divisibility barrier): There is some partition P of V (J) into d ≤ kr parts of size

at least δ∗

to P′.

Note that the fact that the perfect matching in Jkis γ-balanced in the first property is

not stated in the statement of the theorem in [5]. However, examining the short derivation

of this theorem from Theorem 7.11 in [5] shows this to be the case.

k−1(J) − µn such that P refines P′and Lµ

P(Jk) is incomplete with respect

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6PETER KEEVASH AND RICHARD MYCROFT

3. Outline of the proof

V1

V2

V3

V4

?

X2

?

X1

X1

1

X1

2

X1

3

X1

4

X2

1

X2

2

X2

3

X2

4

p1= 2

p2= 1

Figure 2. A row-decomposition of a 4-partite graph G into 2 rows.

In this section we outline the proof of Theorem 1.1. For ease of explanation we restrict to

the case when G is an r-partite graph whose vertex classes each have size kn and δ∗(G) ≥

(k − 1)n. Our strategy consists of the following three steps:

(i) Impose a row structure on G.

(ii) Find balanced perfect clique-packings in each row.

(iii) Glue together the row clique-packings to form a Kk-packing of G.

For step (i) we partition V (G) into blocks Xi

into s blocks X1

j. This partition is best thought of as a s × r grid, with rows

Xi:=?

which satisfies these conditions an s-row-decomposition of G. We also require that G has

density close to 1 between any two blocks which do not lie in the same row or column (we

refer to the smallest such density as the minimum diagonal density). Figure 2 illustrates

this structure. We begin with the trivial 1-row-decomposition of G with a single row (so the

blocks are the vertex classes Vj). If it is possible to split this row into two rows to obtain

a row-decomposition with minimum diagonal density at least 1 − d (where d will be small),

then we say that G is d-splittable. If so, we partition G in this manner, and then examine in

turn whether either of the two rows obtained is splittable (for some larger value of d). By

repeating this process, we obtain a row-decomposition of G with high minimum diagonal

density in which no row is splittable; this argument is formalised in Lemma 4.1.

For step (ii) we require a balanced perfect Kpi-packing in each row Xi. We first use the

results of Section 2 to obtain a near-balanced perfect Kpi-packing in G[Xi]. Fix i and take

J to be the clique pi-complex of G[Xi]. So we regard the row Xias an r-partite vertex set

whose parts are the blocks Xi

j ≤ pi. The assumption δ∗(G) ≥ (k − 1)n implies that

δ∗(J) ≥ (pin,(pi− 1)n,(pi− 2)n,...,n).

Then Theorem 2.3 (with piplaying the role of k) implies that Jpicontains a near-balanced

perfect matching, unless Jpiis close to a space or divisibility barrier. In Section 4 we consider

j, so that each vertex class Vjis partitioned

j,...,Xs

jand columns the vertex classes Vj=?

j∈[r]Xi

i∈[s]Xi

j. We insist that all the blocks

i∈[s]pi= k. We call a partition of V (G) in a given row Xihave equal size pin, where?

1,...,Xi

r, and the edges of Jj are the j-cliques in G[Xi] for

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A MULTIPARTITE HAJNAL-SZEMER´EDI THEOREM7

a space barrier, showing in Lemma 4.2 that since G[Xi] is not d-splittable, Jpicannot be

close to a space barrier. We then consider a divisibility barrier in Section 5. For pi≥ 3,

Lemma 5.3 shows that since G[Xi] is not d-splittable, Jpialso cannot be close to a divisibility

barrier. However, the analogous statement for pi= 2 is false, for the following reason.

We say that G[Xi] is ‘pair-complete’ if it has a structure close to that which appears in

rows V1and V2of Construction 1.2. That is, there is a partition of Xiinto ‘halves’ S

and Xi\ S, such that each vertex class Vj is partitioned into two equal parts, and both

G[S] and G[Xi\ S] are almost complete r-partite graphs. Such a row is not d-splittable if

r is odd, but J2is close to a divisibility barrier. However, Lemma 5.2 shows that this is

essentially the only such example, that is, that if pi= 2 and G[Xi] is neither d-splittable

nor pair-complete then J2is not close to a divisibility barrier. So unless pi= 2 and G[Xi] is

pair-complete, Theorem 2.3 implies that Jpicontains a near-balanced perfect Kpi-packing.

In Section 6 we then show that we can actually obtain a balanced perfect matching in Jpi.

Indeed, in Lemma 6.2 we first delete some ‘configurations’ from G[Xi]; these are subgraphs

of G[Xi] that can be expressed as two disjoint copies of Kpiin G[Xi] in two different ways

(with different index sets). After these deletions we proceed as just described to find a near-

balanced perfect Kpi-packing in G[Xi]. Then by carefully choosing which pair of disjoint

edges to add to the matching from each ‘configuration’, we obtain a balanced perfect Kpi-

packing in G[Xi], as required. This leaves only the case where pi= 2 and G[Xi] is pair-

complete; in this case Lemma 6.4 gives a balanced perfect K2-packing in G[Xi], provided

that each half has even size.

For step (iii), we construct auxiliary hypergraphs, perfect matchings in which describe

how to glue together the perfect Kpi-packings in the rows into a perfect Kk-packing of

G. Recall that the row-decomposition of G was chosen to have large minimum diagonal

density, so almost every vertex of any block Xi

in a different row and column. Assume for now that this row-decomposition of G has the

stronger condition of large minimum diagonal degree, i.e. that we can delete ‘almost’ from

the previous statement. For each row i, we partition its perfect Kpi-packing into sets Eσ,i,

one for each injective function σ : [k] → [r]. For each σ we then form an auxiliary s-partite

s-graph Hσ, where for each i ∈ [s] the i-th vertex class of Hσ is the set Eσ,i (so a copy

of Kpiin G[Xi] is a vertex of Hσ). Edges in Hσ are those s-tuples of vertices for which

the corresponding copies of Kpitogether form a copy of Kkin G. We defer the details of

the partition to the final section of this paper; the crucial point is that the large minimum

diagonal degree of G ensures that each Hσhas sufficiently large vertex degree to guarantee

a perfect matching. Taking the copies of Kk in G corresponding to the union of these

matchings gives a perfect Kk-packing in G, completing the proof.

The above sketch glosses over the use of the precise minimum degree condition in The-

orem 1.1. Indeed, to replace our minimum diagonal density condition with a minimum

diagonal degree condition, we must remove all ‘bad’ vertices, namely those which have many

non-neighbours in some block in a different row and column. To achieve this, before step

(ii) we delete some vertex-disjoint copies of Kkfrom G which cover all bad vertices. We

must ensure that the number of vertices deleted from row Xiis a constant multiple of pifor

each i, so that we will be able to join together the Kpi-packings of the undeleted vertices

of each Xito form a Kk-packing of G. We also need to ensure that each half has even size

in pair-complete rows. This is accomplished in Section 7, which is the most lengthy and

jhas few non-neighbours in any block Xi′

j′

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A MULTIPARTITE HAJNAL-SZEMER´EDI THEOREM37

any j ∈ [r] we have

|V (M4) ∩ Wj| =

?

A∈([r]

k):j∈A

(C′+ N′

A− NA) = C′

?r − 1

k − 1

?

− a′

j=k|M4|

r

− a′

j.

Then |Vj∩?

j ∈ [r]. Finally, |?

The final Kk-packing is chosen so that the remaining blocks all have size proportional to

their row size.

i∈[4]V (Mi)| = k|M4|/r − a′

i∈[4]V (Mi)| ≤ C′?[r]

j+ |?

i∈[3]V (Mi) ∩ Vj| = k|?

i∈[4]Mi|/r for any

k

?+ 2βrk3n ≤ β′n/2.

?

Claim 7.10. (Balancing blocks) There is a Kk-packing M5vertex-disjoint from?

V (M), X′i=?

X′i

has at most βn ≤ αn′non-neighbours in any X′i′

Proof. For each i ∈ [s] and j ∈ [r] we let W′i

W′

j:=?

properly-distributed. Recall also that |W′

Q = (qij) be the s by r integer matrix whose (i,j) entry is qij= |W′i

row of Q sums to zero. Furthermore, since |W′

?

i.e. each column of Q also sums to zero. We also have?

We write Q =?

or (i,j) = (c,d), equal to −1 if (i,j) = (a,d) or (i,j) = (c,b), and equal to zero otherwise.

To see that such a representation is possible, we repeat the following step. Suppose qab> 0

for some a,b. Since each row and column of Q sum to zero, we may choose c,d such that

qad< 0 and qcb< 0. Then Q′:= Q−Qabcdis an s by k integer matrix in which the entries of

each row and column sum to zero. Also, writing Q′= (q′

By iterating this process at most?

Let P denote the set of all families A of pairwise vertex-disjoint subsets Ai⊆ [r] with

|Ai| = pifor i ∈ [s]. To implement a matrix Qabcd∈ Q we fix any two families A,A′such

that b ∈ Aaand d ∈ Ac, and A′is formed from A by swapping b and d. That is, A′

i ∈ [s]\{a,c}, A′a= (Aa\{b})∪{d} and A′c= (Ac\{d})∪{b}. For each A ∈ P let QAbe the

number of times it is chosen as A for some Qabcd, and Q′

A′for some Qabcd. Fix an integer C′′such that kr·r! divides C′′and rkβ′n ≤ C′′≤ 2rkβ′n.

For each A ∈ P let NA= C′′+ QA− Q′

i∈[4]Mi,

j:= Wi

consisting of properly-distributed cliques, such that defining M =?

i ∈ [s] and j ∈ [r], where n′= |X′|/kr is an integer divisible by r! with n′≥ n−ζn/2. Thus

jforms a row-decomposition of G′:= G[X′] of type p. Furthermore, any vertex v ∈ X′i

i∈[5]Mi, X′i

j| = pin′for any

j\

j∈[r]X′i

jand X′=?

i∈[s]X′i= V (G) \ V (M), we have |X′i

j

j′ with i′?= i.

j= Wi

j\ V (?

i∈[4]Mi), W′i:=?

j∈[r]W′i

j,

i∈[s]W′i

j, and W′=?

j∈[r]W′

j. We may fix an integer D so that |W′i| = piD for

any i ∈ [s], since |Wi\V (M1∪M2)|/piis constant for i ∈ [s] and each clique in M3∪ M4is

j| is constant for each j ∈ [r] by choice of M4. Let

j| − piD/r. Then each

j| is constant for each j ∈ [r] we have

j| − piD/r) = |W′

i∈[s]

qij=

?

i∈[s]

(|W′i

j| − |W′|/r = 0,

i,j|qij| ≤ β′n using (A2) and Claim

7.9(ii).

A∈QA, where Q is a multiset of matrices. Each matrix in Q is of the form

Qabcd, for some a,c ∈ [s] and b,d ∈ [r], defined to have (i,j) entry equal to 1 if (i,j) = (a,b)

ij), we have?

i,j|q′

ij| ≤?

i,j|qij|−2.

i,j|qij|/2 times we obtain the all-zero matrix, whereupon

we have expressed Q in the required form with |Q| ≤ β′n, counting with multiplicity.

i= Aiif

Athe number of times it is chosen as

A, so NA≥ 0.

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38PETER KEEVASH AND RICHARD MYCROFT

Now we greedily choose M5 to consist of NA copies of K′for each A ∈ P, each of

which will intersect each Wiin precisely those Wi

|M5| =?

there are at most kζn such vertices. By Claim 7.5(i), for any A ∈ P there are at least ωnk

properly-distributed copies of Kkin G which intersect each Wiin precisely those Wi

that j ∈ Ai, so we can indeed choose M5greedily. This defines M, X′i

as in the statement of the claim. Note that |M| ≤ ζn, and kr · r! divides |X′| by Claims

7.8(ii) and 7.9(iii) and the choice of C′′.

Finally, consider the number of vertices used in Wa

chosen uniformly at random, then Aais a uniformly random subset of size pa, so contains b

with probability pa/r. So if we chose C′′copies of each A ∈ P we would choose paN vertices

in Wa

to adjust by QA− Q′

more vertex in each of Wa

Q =?

r!. Note also that n − n′≤ |V (M)|/kr ≤ ζn/2. Lastly, by choice of M3every bad vertex

is covered by M, so any vertex v ∈ X′i

with i′?= i and j′?= j by (A3).

jsuch that j ∈ Ai. Then we will have

A∈PNA = C′′|P| ≤ ζn/2. When choosing any copy of Kkwe must avoid the

vertices of copies of Kk which were previously chosen for M5, or which lie in?

i∈[4]Mi;

jsuch

j, X′i, X′

j, X′, n′, G′

b, where a ∈ [s], b ∈ [r]. If A ∈ P is

b, where N := C′′|P|/r. However, since we choose NAcopies of each A ∈ P, we need

A. These are chosen so that for each matrix Qabcd∈ Q we choose one

band Wc

A∈QA we thus use paN +qabvertices in Wa

pa(D/r − N). Writing n′= D/r − N, we have n′= |X′|/kr, so n′is an integer divisible by

d, and one fewer vertex in each of Wa

b. This gives |X′a

dand Wc

b|−paN −qab=

b. Since

b| = |W′a

jhas at most βn ≤ αn′non-neighbours in any X′i′

j′

?

After deleting M as in Claim 7.10, we obtain G′with an s-row-decomposition X′that

satisfies conclusion (i) of Lemma 7.2. To complete the proof, we need to satisfy conclusion

(ii), by finding a balanced perfect Kpi-packing in row i for each i ∈ [s]. We consider two

cases according to whether or not there are multiple rows with pi≥ 2.

Case 1: There is at most one row i with pi≥ 2.

Case 1.1: pi = 1 for any i ∈ [s]. For each i ∈ [s] there is a trivial balanced perfect

K1-packing in G[X′i], namely {{v} : v ∈ X′i}. This satisfies condition (ii), so the proof is

complete in this case.

Case 1.2: G has the extremal row structure.

complete row i∗, and pi= 1 for any i ?= i∗. There is a trivial balanced perfect K1-packing

in G[X′i] for any i ?= i∗, so it remains only to find a balanced perfect matching in G[X′i∗].

Since row i∗is pair-complete, we chose sets Si∗

S′i∗

j\ V (M) for each j ∈ [r]. Then

This means that G has one pair-

jfor j ∈ [r] when forming the sets Wi

j. Let

j:= Si∗

j∩ X′i

j= Si∗

(1 + β)n

(A2)

≥ |Wi∗

j| = (1 ± 2ζ)n′for each j ∈ [r]. Recall also that in this case we required that

|Si∗\ V (M1)| was even when choosing M1, and M2was empty. Furthermore, any copy K′

of Kkin M3∪ M4∪ M5was chosen to be properly-distributed, so in particular |K′∩ Si∗| is

even. We conclude that |Si∗\ V (M)| is even. Since every bad vertex of G was covered by

M3⊆ M, by (A3) for any j′?= j any vertex in S′i∗

in S′i∗

j| − |Yi∗

j \ Si∗

j| ≥ |Si∗

j| ≥ |S′i∗

j| ≥ |Si∗

j| − |M|

(A2)

≥ (1 − β/2)n − ζn,

and so |S′i∗

jhas at most βn ≤ 2ζn′non-neighbours

jhas at most βn ≤ 2ζn′non-neighbours in X′i∗

j′, and any vertex in X′i∗

j\S′i∗

j′ \S′i∗

j′ by

Page 39

A MULTIPARTITE HAJNAL-SZEMER´EDI THEOREM39

(A3). By Lemma 6.4 (with 2ζ in place of ζ) we conclude that G[X′i∗] contains a balanced

perfect matching, completing the proof in this case.

Case 1.3: pℓ= 2 for some ℓ, pi′ = 1 for any i′?= ℓ, but row ℓ is not pair-complete.

Recall that this means that G1[Xℓ] was not d′-pair-complete. Recall also that G1[Xℓ] was not

d′-splittable (this is true of any row of G1). Since by (A2) we have |Xℓ

any j ∈ [r], Proposition 6.1 implies that G[X′ℓ] is neither d′′-splittable nor d′′-pair-complete.

So G[X′ℓ] contains a ν-balanced perfect matching by Corollary 5.7. Then Proposition 6.5

implies that there exists an integer D ≤ 2νn′and a Kk-packing M∗in G such that r! divides

n′′:= n′−D, M∗covers piD vertices in X′i

contains a balanced perfect matching. Note that since n′≥ n−ζn/2 we have n′′≥ n+/k−ζn.

We add the copies of Kkin M∗to M, and let X′′i

X′i, X′, G′by deleting the vertices covered by M∗. This leaves an s-row-decomposition of

G′′into blocks X′′i

before G[X′′i] contains a trivial balanced K1-packing for every i ?= ℓ. Finally, any vertex

v ∈ X′i

has at least pi′n′− 2βn ≥ pi′n′′− αn′′. So the enlarged matching M, restricted blocks X′′i

and G′′satisfy (i) and (ii) with n′′in place of n′, which completes the proof of this case, and

so of Case 1.

j△X′ℓ

j| ≤ 2ζ(2n) for

jfor any i ∈ [s] and j ∈ [r], and G[X′ℓ\V (M∗)]

j, X′′i, X′′, G′′be obtained from X′i

j,

jof size pin′′, in which G[X′′ℓ] contains a balanced perfect matching. As

jhas lost at most k|M∗| ≤ βn neighbours in X′i′

j′ for any i ?= i′and j ?= j′, so still

j

Case 2: There are at least two rows i with pi≥ 2. In this case we modify the cliques in

M2and the blocks X′i

Kpi-packing for each i ∈ [s]. We proceed through each i ∈ [s] in turn. When considering

any i ∈ [s] we leave all blocks X′i′

that G[X′i

j] contains a trivial balanced K1-packing. Suppose next that pi≥ 3, and recall

that G1[X1] was not d′-splittable. As in Case 1.3, Proposition 6.1 implies that G[X′i] is not

d′′-splittable. So G[X′i] contains a balanced perfect Kpi-packing by Lemma 6.2.

This leaves only those rows i ∈ [s] with pi= 2 to consider. Suppose first that row i is pair-

complete. As in Case 1.2 we let S′i

for each j ∈ [r], and for any j′?= j, any vertex in S′i

S′i

even then G[X′i

j] contains a balanced perfect matching by Lemma 6.4. So we may suppose

that |S′i| is odd. Fix any i′with i′?= i and pi′ ≥ 2. We choose any i′i-distributed copy K′of

Kkin M2, let x be the vertex of K′in X′i, and let j be such that x ∈ X′i

every vertex in K′is good, so at least 2n′− kβn vertices y ∈ X′i

member of V (K′) \ {x}. So we may choose a vertex y ∈ X′i

|{x,y}∩Si

and replace y by x in X′i

x ∈ Si

to find a balanced perfect matching in G[X′i].

Finally suppose that pi= 2 and row i is not pair-complete, that is G1[Xi] was not d′-

pair-complete. Since G1[Xi] was also not d′-splittable, as before Proposition 6.1 implies that

G[X′i] is neither d′′-splittable nor d′′-pair-complete. Fix any i′with i′?= i and pi′ ≥ 2. Recall

that M2contains ⌈ηn⌉ copies K′of Kkin G[Y ] which are i′i-distributed. We can fix q ∈ [r]

so that at least ηn/r such K′have exactly one vertex in Yi

jso that after these modifications G[X′i

j] contains a balanced perfect

jwith i′?= i unaltered. If pi= 1, then we have already seen

j= Si

j∩X′i

jfor each j ∈ [r], which gives |S′i

jhas at most 2ζn′non-neighbours in

jhas at most 2ζn′non-neighbours in X′i

j| = (1+2ζ)n′

j′, and any vertex in X′i

j\ S′i

j′ \ S′i

j′. If |S′i| is

j. By Claim 7.7(i),

jare adjacent to every

v∈V (K′)\{x}N(v) such that

j∩?

jif y ∈ S′i

j| is odd. We replace K′in M by the copy of Kkin G induced by {y}∪V (K′)\{x},

jand G′. We also delete y from S′i

j. So |S′i

j, and add x to S′i

jif

j| is now even. Since x is good, we may now apply Lemma 6.4 as in Case 1.2

q. We assign arbitrarily ηn/r such

Page 40

40PETER KEEVASH AND RICHARD MYCROFT

K′to each ordered triple (i1,i2,i3) of distinct elements of [r] \ {q}, so that at least ηn/r4

of the K′are assigned to each triple. Now, fix any triple (j1,j2,j3) and any K′which was

assigned to it. Let x be the vertex of K′in Yi

q, and consider paths xx1x2x3y of length 4 in

G with xℓ∈ X′i

is good and does not lie in Wi, at most kβn vertices y ∈ X′i

V (K′)\{x}. Choosing x1,x2,x3and y in turn, recalling (†) and n′≥ n−ζn/2, there are at

least 2n′−n ≥ (1−ζ)n′choices for each xℓ, and at least 2n′−n−(k−1)βn ≥ (1−ζ)n′choices

for y. We obtain at least n4/2 such paths, and so we may fix some y = y(x) which lies in at

least n3/5 such paths. For each of these n3/5 paths xx1x2x3y we add a ‘fake’ edge between

y and x1. Then allowing the use of fake edges, y lies in at least n3/5 4-cycles x1x2x3y of

length 4 in G with xℓ∈ X′i

K′in M2which was assigned to the triple (j1,j2,j3), for every ordered triple (j1,j2,j3) of

distinct elements of [r]\{q}. Let G∗be the graph formed from G[X′i] by the addition of fake

edges. Then by construction, for any triple (j1,j2,j3) there are at least (ηn/r4)(n3/5) ≥ νn4

4-cycles yx1x2x3in G∗with y ∈ X′i

a spanning subgraph G[X′i] which is neither d′′-splittable nor d′′-pair-complete, and has

δ∗(G[X′i]) ≥ 2n′− n ≥ n′− ζn. Then G∗contains a balanced perfect matching M∗by

Lemma 6.2 (with q in place of 1). Of course, M∗may contain fake edges. However, any

fake edge in M∗is of the form y(x)x1, where x1is a neighbour of x, and x lies in some K′

in M2. Since y(x) is uniquely determined by x and M∗is a matching, at most one edge in

M∗has the form y(x)x1for each x. Furthermore, by choice of y = y(x), {y} ∪ V (K′) \ {x}

induces a copy K′

replace y in X′i

G. We carry out these substitutions for every fake edge in M∗, at the end of which M∗is a

perfect matching in G[X′i], which is balanced since each edge was replaced with another of

the same index.

When considering row i we only replace cliques of M2that are i′i-distributed for some

i′?= i. These cliques uniquely determine i, so do not affect the replacements for other rows.

We may therefore proceed through every i ∈ [s] in this manner. After doing so, the modified

blocks X′i

this row-decomposition of the modified G′satisfies condition (ii) of the lemma. Note that

we still have |X′i

vertices, every vertex in any modified block X′i

holds as in Claim 7.10. This completes the proof of Lemma 7.2.

jℓfor ℓ ∈ [3] and y ∈ X′i

q∩?

v∈V (K′)\{x}N(v). Since every vertex of K′\{x}

qfail to be adjacent to all of

jℓfor ℓ ∈ [3]. We introduce fake edges in this manner for every

qand xℓ∈ X′i

jℓfor ℓ ∈ [3]. Furthermore, G∗contains

yof Kkin G, and xx1is an edge. So we may replace K′in M2by K′

qby x (note that x is good), and the fake edge yx1in M∗by the edge xx1of

y,

jare such that G[X′i] contains a balanced perfect Kpi-packing for any i ∈ [s], i.e.

j| = pin′for any i ∈ [s]. Since we only replaced good vertices with good

jis good, and so condition (i) of the lemma

8. Completing the proof of Theorem 1.1.

In this final section we combine the results of previous sections to prove Theorem 1.1. We

also need the following lemma from [5], which gives a minimum degree condition for finding

a perfect matching in a k-partite k-graph whose vertex classes each have size n.

Lemma 8.1 ([5]). Suppose 1/n ≪ dk≪ 1/k and G is a k-partite k-graph on vertex classes

V1,...,Vkof size n. If every vertex of G lies in at least (1 − dk)nk−1edges of G then G

contains a perfect matching.

We can now give the proof of Theorem 1.1, as outlined in Section 3, which we first restate.

Page 41

A MULTIPARTITE HAJNAL-SZEMER´EDI THEOREM 41

Theorem 1.1.

following statement holds. Let G be a r-partite graph whose vertex classes each have size n

such that δ∗(G) ≥ (k − 1)n/k. Then G contains a perfect Kk-packing, unless rn/k is odd,

k | n, and G∼= Γn,r,k.

Proof. First suppose that k = 2, so a perfect Kk-packing is a perfect matching. If r = 2

then G is a bipartite graph with minimum degree at least n/2, so has a perfect matching.

For r ≥ 3 the result follows from Tutte’s theorem, which states that a graph G on the

vertex set V contains a perfect matching if and only if for any U ⊆ V the number of odd

components (i.e. connected components of odd size) in G[V \U] is at most |U|. To see that

this implies the theorem for k = 2, suppose for a contradiction that there is some U ⊆ V

for which G[V \ U] has more than |U| odd components. Clearly |U| < |V |/2 = rn/2. So by

averaging U has fewer than n/2 vertices in some Vj. Since δ∗(G) ≥ n/2, every v ∈ V \Vjhas

a neighbour in Vj\ U, so G[V \ U] has at most |Vj\ U| ≤ n components. So we must have

|U| < n. But then U must have fewer than n/r ≤ n/3 vertices in some Vj, so any v ∈ V \Vj

has more than n/6 neighbours in Vj. It follows that G[V \ U] has at most 5 components,

so |U| < 5. So any v ∈ V \ Vjactually has more than n/2 − 5 > n/3 neighbours in Vj, so

G[V \U] has at most 2 components. So |U| ≤ 1. If |U| = 1 then |V \U| is odd, so G[V \U]

cannot have 2 odd components. The only remaining possibility is that |U| = 0 and G has 2

odd components. Let C1and C2be these components, and for each i ∈ [2] and j ∈ [r] let Vi

be the vertices of Vjcovered by Ci. Then |Vi

and G[Vi

odd, 2 divides n and G is isomorphic to Γn,r,2, contradicting our assumption.

We may therefore assume that k ≥ 3. If r = k = 3 then Theorem 1.1 holds by the

result of [8]. So we may assume that r > 3. We introduce new constants d and d′with

1/n ≪ d ≪ d′≪ 1/r. Since r > 3 and r ≥ k ≥ 3 we may apply Lemma 7.2 (with n and d

in place of n+and α) to delete the vertices of a collection of pairwise vertex-disjoint copies

of Kkfrom G. Letting V′be the set of undeleted vertices, we obtain G′= G[V′] and an

s-row-decomposition of G′into blocks X′i

and pi∈ [k] with?

(i) for each i,i′∈ [s] with i ?= i′and j,j′∈ [r] with j ?= j′, any vertex v ∈ X′i

pi′n′− dn′neighbours in X′i′

(ii) for every i ∈ [s] the row G[X′i] contains a balanced perfect Kpi-packing Mi.

Note that we must have |Mi| = rn′for any i ∈ [s].

Now we implement step (iii) of the proof outline, by constructing auxiliary hypergraphs,

perfect matchings of which describe how to glue together the perfect Kpi-packings in the

rows into a perfect Kk-packing of G. We partition [k] arbitrarily into sets Aiwith |Ai| = pi

for i ∈ [s]. Let Σ denote the set of all injective functions σ : [k] → [r]. For each i ∈ [s] we

partition Miinto sets Ei

r!

index σ(Ai). To see that this is possible, fix any B ∈?[r]

Since

(r−k)!

pi

?

{e1,e2,...,es} with ei∈ Ei

For any r ≥ k there exists n0such that for any n ≥ n0with k | rn the

j

j| ≥ δ∗(G) ≥ n/2, so we deduce that |Vi

j| = n/2

j,Vi

j′] is a complete bipartite graph for any i ∈ [2] and j ?= j′. So |C1| = rn/2 is

jof size pin′for i ∈ [s] and j ∈ [r], for some s ∈ [k]

i∈[s]pi= k, such that r! | n′, n′≥ n/k − dn and

jhas at least

j′, and

σof size N :=rn′(r−k)!

for σ ∈ Σ, so that each member of Ei

pi

members of Σ with σ(Ai) = B.

we may choose the sets Ei

σas required. For every σ ∈ Σ,

σhas

?. Since Miis balanced, rn′/?r

(r−k)!

pi

?

members of Mihave index B. Note that there arepi!(r−pi)!

· N = rn′/?r

σfor each i ∈ [s] is an edge of Hσif and only if xy ∈ G for any

pi!(r−pi)!

we form an auxiliary s-partite s-graph Hσ with vertex classes Ei

σfor i ∈ [s], where a set

Page 42

42PETER KEEVASH AND RICHARD MYCROFT

i ?= j, x ∈ eiand y ∈ ej. Thus Hσhas N vertices in each vertex class, and e1e2...esis an

edge of Hσif and only if?

for any ei∈ Ei

x ∈ V (ei) has at most dn′non-neighbours in each X′ℓ

most pidn′vertices of X′ℓ

jare not neighbours of some vertex of ei. Now we can estimate the

number of (s − 1)-tuples (ej∈ Ej

There are fewer than k choices for j ∈ [s]\{i}, at most pidn′elements ej∈ Ej

a non-neighbour of some vertex of ei, and at most Ns−2choices for ej′ ∈ Ej′

So the number of edges of Hσcontaining eiis at least Ns−1− kpidn′Ns−2≥ (1 − d′)Ns−1.

Since eiwas arbitrary, Lemma 8.1 gives a perfect matching in each Hσ. This corresponds to

a perfect Kk-packing in G covering the vertices of?

pairwise vertex-disjoint copies of Kkdeleted in forming G′, we obtain a perfect Kk-packing

in G.

j∈[s]V (ej) induces a copy of Kkin G.

Next we show that each Hσ has high minimum degree. Fix σ ∈ Σ and i ∈ [s]. Then

σ, eiis a copy of Kpiin G[X′i] with index σ(Ai), and so by (i) each vertex

jwith ℓ ?= i and j / ∈ σ(Ai). So at

σ: j ∈ [s] \ {i}) so that {e1,...,es} is not an edge of Hσ.

σthat contain

σ, j′∈ [s]\{i,j}.

i∈[s]V (Ei

σ), where V (Ei

σ) denotes the

vertices covered by members of Ei

σ. Combining these perfect Kk-packings, and adding the

?

9. Concluding remarks

By examining the proof, one can obtain a partial stability result, i.e. some approximate

structure for any r-partite graph G with vertex classes each of size n, where k | rn, such

that δ∗(G) ≥ (k − 1)n/k − o(n), but G does not contain a perfect Kk-packing. To do this,

note that under this weaker minimum degree assumption, the n in (†) must replaced by

n+o(n). We now say that a block Xi

jis bad with respect to v if v has more than n/2+o(n)

non-neighbours in Xi

j, so it is still true that at most one block in each column is bad with

respect to a given vertex. Then each of our applications of (†) proceeds as before, except

for in Claim 7.6, where we used the exact statement of (†) (i.e. the exact minimum degree

hypothesis). This was needed to choose a matching Eiin G[Wi] of size ai, each of whose

edges contains a good vertex, for each i ∈ I+with pi= 1. If we can choose such matchings

Eithen the rest of the proof to give a perfect Kk-packing still works under the assumption

δ∗(G) ≥ (k − 1)n/k − o(n). So we can assume that there is some i ∈ I+with pi= 1 for

which no such matching exists. Since the number of bad vertices and aiare o(n), it follows

that Wiis a subset of size rn/k + o(n) with o(n2) edges, i.e. we have a sparse set of about

1/k-proportion of the vertices. On the other hand, this is essentially all that can be said

about the structure of G, as any such G with an independent set of size rn/k + 1 cannot

have a perfect Kk-packing.

References

[1] B. Csaba and M. Mydlarz, Approximate Multipartite Version of the Hajnal–Szemer´ edi Theorem, J.

Combinatorial Theory, Series B, to appear.

[2] E. Fischer, Variants of the Hajnal-Szemer´ edi Theorem, J. Graph Theory 31 (1999), 275–282.

[3] S. Hakimi, On the realizability of a set of integers as degrees of the vertices of a graph, SIAM J. Appl.

Math. 10 (1962), 496–506.

[4] S. Janson, T. ? Luczak and A. Ruci´ nski, Random graphs, Wiley-Interscience, 2000.

[5] P. Keevash and R. Mycroft, A geometric theory for hypergraph matching, arXiv:1108.1757.

[6] D. K¨ uhn and D. Osthus, Embedding large subgraphs into dense graphs, Surveys in Combinatorics 2009,

Cambridge University Press, 2009, 137–167.

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A MULTIPARTITE HAJNAL-SZEMER´EDI THEOREM43

[7] A. Lo and K. Markstr¨ om, A multipartite version of the Hajnal-Szemer´ edi theorem for graphs and

hypergraphs, arXiv:1108.4184.

[8] C. Magyar and R. Martin, Tripartite version of the Corr´ adi-Hajnal theorem, Disc. Math. 254 (2002),

289–308.

[9] R. Martin and E. Szemer´ edi, Quadripartite version of the Hajnal-Szemer´ edi theorem, Disc. Math. 308

(2008), 4337–4360.

Peter Keevash

School of Mathematical Sciences

Queen Mary, University of London

London, UK

Richard Mycroft

School of Mathematics

University of Birmingham

Birmingham, UK

E-mail addresses: p.keevash@qmul.ac.uk, r.mycroft@bham.ac.uk