Equitable list coloring and treewidth

Abstract

Abstract Given 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 ≥3 w −1. We also show that if G has treewidth w ≥5, then G is equitably k ‐choosable for k ≥max{Δ( G )+ w −4, 3 w −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