Properly 2-Colouring Linear Hypergraphs

Abstract

Using the symmetric form of the Lovász Local Lemma, one can conclude that a k-uniform hypergraph $\mathcal{H}$ admits a proper 2-colouring if the maximum degree (denoted by Δ) of $\mathcal{H}$ is at most $\frac{2^k}{8k}$ independently of the size of the hypergraph. However, this argument does not give us an algorithm to find a proper 2-colouring of such hypergraphs. We call a hypergraph linear if no two hyperedges have more than one vertex in common.

In this paper, we present a deterministic polynomial time algorithm for 2-colouring every k-uniform linear hypergraph with $\Delta \le 2^{k-k^{\epsilon}}$, where 1/2 < ε< 1 is any arbitrary constant and k is larger than a certain constant that depends on ε. The previous best algorithm for 2-colouring linear hypergraphs is due to Beck and Lodha [4]. They showed that for every δ> 0 there exists a c > 0 such that every linear hypergraph with Δ ≤ 2^{k − δk} and $k > c\log\log(|E(\mathcal{H})|)$, can be properly 2-coloured deterministically in polynomial time.

Research of the first author is supported by a NSERC graduate scholarship and research grants of Prof. D. Thérien, the second author is supported by a Canada Research Chair in graph theory. We would like to thank an anonymous referee for pointing out the reference [10].