Toward a Hajnal-Szemeredi theorem for hypergraphs

Details Toward a Hajnal-Szemeredi theorem for hypergraphs Toward a Hajnal-Szemeredi theorem for hypergraphs

2010-05-21

arxiv.org/abs/1005.4079v1

Mathematics

Combinatorics

Scientific paper

Let $H$ be a triple system with maximum degree $d>1$ and let $r>10^7\sqrt{d}\log^{2}d$. Then $H$ has a proper vertex coloring with $r$ colors such that any two color classes differ in size by at most one. The bound on $r$ is sharp in order of magnitude apart from the logarithmic factors. Moreover, such an $r$-coloring can be found via a randomized algorithm whose expected running time is polynomial in the number of vertices of $\cH$. This is the first result in the direction of generalizing the Hajnal-Szemer\'edi theorem to hypergraphs.

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