Toward a Hajnal-Szemeredi theorem for hypergraphs

TL;DR

This paper extends the Hajnal-Szemerédi theorem to hypergraphs by establishing a proper vertex coloring with nearly equal class sizes for certain triple systems, using a polynomial-time randomized algorithm.

Contribution

It provides the first hypergraph generalization of the Hajnal-Szemerédi theorem with explicit bounds and an efficient coloring algorithm.

Findings

Established a proper vertex coloring with r colors for hypergraphs with maximum degree d.

Proved the bound on r is sharp up to logarithmic factors.

Developed a polynomial-time randomized algorithm for finding such colorings.

Abstract

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.