On (4, 2)‐Choosable Graphs

Abstract

AbstractA graph G is called ‐choosable if for any list assignment L that assigns to each vertex v a set of a permissible colors, there is a b‐tuple L‐coloring of G. An (a, 1)‐choosable graph is also called a‐choosable. In the pioneering article on list coloring of graphs by Erdős et al. , 2‐choosable graphs are characterized. Confirming a special case of a conjecture in , Tuza and Voigt proved that 2‐choosable graphs are ‐choosable for any positive integer m. On the other hand, Voigt proved that if m is an odd integer, then these are the only ‐choosable graphs; however, when m is even, there are ‐choosable graphs that are not 2‐choosable. A graph is called 3‐choosable‐critical if it is not 2‐choosable, but all its proper subgraphs are 2‐choosable. Voigt conjectured that for every positive integer m, all bipartite 3‐choosable‐critical graphs are ‐choosable. In this article, we determine which 3‐choosable‐critical graphs are (4, 2)‐choosable, refuting Voigt's conjecture in the process. Nevertheless, a weaker version of the conjecture is true: we prove that there is an even integer k such that for any positive integer m, every bipartite 3‐choosable‐critical graph is ‐choosable. Moving beyond 3‐choosable‐critical graphs, we present an infinite family of non‐3‐choosable‐critical graphs that have been shown by computer analysis to be (4, 2)‐choosable. This shows that the family of all (4, 2)‐choosable graphs has rich structure.