List Coloring Triangle-Free Hypergraphs

Abstract

A triangle in a hypergraph is a collection of distinct vertices u,v,w and distinct edges e,f,g with u,v \in e, v,w \in f, w,u \in g, and \{u,v,w\} \cap e \cap f \cap g=\emptyset. The i-degree of a vertex in a hypergraph is the number of edges of size i containing it. We prove that every triangle-free hypergraph of rank three (edges have size two or three) with maximum 3-degree \Delta_3 and maximum 2-degree \Delta_2 has list chromatic number at most c max{\Delta_2/ log{\Delta_2}}, (\Delta_3 / log{\Delta_3})^(1/2)} for some absolute positive constant c. This generalizes a result of Johansson and a result of Frieze and the second author.