Equitable list coloring and treewidth

Abstract

AbstractGiven lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most ⌈|V(G)|/k⌉ vertices. A graph is equitably k ‐choosable if such a coloring exists whenever the lists all have size k. Kostochka, Pelsmajer, and West introduced this notion and conjectured that G is equitably k‐choosable for k>Δ(G). We prove this for graphs of treewidth w≤5 if also k≥3w−1. We also show that if G has treewidth w≥5, then G is equitably k‐choosable for k≥max{Δ(G)+w−4, 3w−1}. As a corollary, if G is chordal, then G is equitably k‐choosable for k≥3Δ(G)−4 when Δ(G)>2. © 2009 Wiley Periodicals, Inc. J Graph Theory