Near-optimal list colorings

We show that a simple variant of a naive colouring procedure can be used to list colour the edges of a k-uniform linear hypergraph of maximum degree provided every vertex has a list of at least +c(log) 4 1? 1 k available colours (where c is a constant which depends on k). We can extend this to colour hypergraphs of maximum codegree o(() with + o(() colours. This improves earlier results of Kahn and our techniques are quite similar. We also develop eecient algorithms to obtain such colourings when is constant. A hypergraph H consists of a set V (H) (or simply V) of vertices and a set E(H) (or simply E) of edges, each of which is a subset of V (H). The chromatic index of a hypergraph is the minimum number of colours needed to colour its edges so that no two edges which intersect receive the same colour. Given a list of colours for each edge of H, we say that an edge colouring is acceptable if every edge is coloured with a colour on its list. The list chromatic index of a hypergraph H is the minimum r which satisses: if every list has at least r members then there is an acceptable colouring. Obviously, both of these numbers can be no smaller than the maximum degree of H, denoted (H). A hypergraph is k-uniform if every edge contains k vertices. It is k-bounded if every edge contains at most k vertices. We shall discuss list colouring the edges of k-bounded hypergraphs. For ease of exposition we consider only k-uniform