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# A Multipartite Hajnal-Szemer\'edi Theorem

• ##### Richard Mycroft
01/2012;
Source: arXiv

ABSTRACT 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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##### Article: A multipartite version of the Hajnal-Szemer\'edi theorem for graphs and hypergraphs
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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.
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