A Multipartite Hajnal-Szemer\'edi Theorem

Source: arXiv


The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree
threshold that forces a graph to contain a perfect K_k-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.

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    ABSTRACT: A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemer\'edi states that if $G$ is a graph of order $n$ with minimum degree $\delta(G) \ge (t-1)n/t$ and $t| n$, then $G$ contains a perfect $K_t$-matching. Let $G$ be a $t$-partite graph with vertex classes $V_1$,..., $V_t$ each of size $n$. We show that if every vertex $x \in V_i$ is joined to at least $((t-1)/t + \gamma)n $ vertices of $V_j$ for $i \ne j$, then $G$ contains a perfect $K_t$-matching, thus verifying a conjecture of Fisher asymptotically. Furthermore, we consider a generalisation to hypergraphs in terms of the codegree.
    Combinatorics Probability and Computing 08/2011; 22(1). DOI:10.1017/S096354831200048X · 0.62 Impact Factor
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    ABSTRACT: Let $G$ be a $k$-partite graph with $n$ vertices in parts such that each vertex is adjacent to at least $\delta^*(G)$ vertices in each of the other parts. Magyar and Martin \cite{MaMa} proved that for $k=3$, if $\delta^*(G)\ge 2/3n $ and $n$ is sufficiently large, then $G$ contains a $K_3$-factor (a spanning subgraph consisting of $n$ vertex-disjoint copies of $K_3$) except that $G$ is one particular graph. Martin and Szemer\'edi \cite{MaSz} proved that $G$ contains a $K_4$-factor when $\delta^*(G)\ge 3/4n$ and $n$ is sufficiently large. Both results were proved by the Regularity Lemma. In this paper we give a proof of these two results by the absorbing method. Our absorbing lemma actually works for all $k\ge 3$.


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