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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.

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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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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