Flexible list colorings: Maximizing the number of requests satisfied

Abstract

Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose , is a graph, is a list assignment for , and is a function with nonempty domain such that for each ( is called a request of ). The triple is ‐satisfiable if there exists a proper ‐coloring of such that for at least vertices in . We say is ‐flexible if is ‐satisfiable whenever is a ‐assignment for and is a request of . It was shown by Dvořák et al. that if is prime, is a ‐degenerate graph, and is a request for with domain of size 1, then is 1‐satisfiable whenever is a ‐assignment. In this paper, we extend this result to all for bipartite ‐degenerate graphs.The literature on flexible list coloring tends to focus on showing that for a fixed graph and there exists an such that is ‐flexible, but it is natural to try to find the largest possible for which is ‐flexible. In this vein, we improve a result of Dvořák et al., by showing ‐degenerate graphs are ‐flexible. In pursuit of the largest for which a graph is ‐flexible, we observe that a graph is not ‐flexible for any if and only if , where is the Hall ratio of , and we initiate the study of the list flexibility number of a graph , which is the smallest such that is ‐flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.