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University of Birmingham A Multipartite Hajnal-Szemerédi Theorem

. 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.