Coloring Graphs From Random Lists

Abstract

ABSTRACT Given positive integers and a graph , a family of lists is said to be a random ‐list‐assignment if for every the list is a subset of of size , chosen uniformly at random and independently of the choices of all other vertices. An ‐vertex graph is said to be a.a.s. ‐colorable if , where is a random ‐list‐assignment. We prove that if and , where is the maximum degree of and is an integer, then is a.a.s. ‐colorable. This is not far from being the best possible, forms a continuation of the so‐called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Furthermore, we consider this problem under the additional assumption that is ‐free for some graph . For various graphs , we estimate the smallest for which any ‐free ‐vertex graph is a.a.s. ‐colorable for every . This extends and improves several results of Casselgren.