Properly 2-Colouring Linear Hypergraphs

Using the symmetric form of the Lovász Local Lemma, one can conclude that a k-uniform hypergraph H admits a proper 2-colouring if the maximum degree (denoted by ∆) of H is at most 2 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 ∆ ≤ 2k−k , where 1/2 < < 1 is any arbitrary constant and k is larger than a certain constant that depends on . 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 hypergaph with ∆ ≤ 2k−δk and k > c log log(|E(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