List Coloring a Cartesian Product with a Complete Bipartite Factor

Abstract

We study the list chromatic number of the Cartesian product of any graph G and a complete bipartite graph with partite sets of size a and b, denoted \(\chi _\ell (G \square K_{a,b})\). We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us \(\chi _\ell (K_{a,b}) = 1 + a\) if and only if \(b \ge a^a\). Since \(\chi _\ell (K_{a,b}) \le 1 + a\) for any \(b \in {\mathbb {N}}\), this result tells us the values of b for which \(\chi _\ell (K_{a,b})\) is as large as possible and far from \(\chi (K_{a,b})=2\). In this paper we seek to understand when \(\chi _\ell (G \square K_{a,b})\) is far from \(\chi (G \square K_{a,b}) = \max \{\chi (G), 2 \}\). It is easy to show \(\chi _\ell (G \square K_{a,b}) \le \chi _\ell (G) + a\). Borowiecki et al. (Discrete Math 306:1955–1958, 2006) showed that this bound is attainable if b is sufficiently large; specifically, \(\chi _\ell (G \square K_{a,b}) = \chi _\ell (G) + a\) whenever \(b \ge (\chi _\ell (G) + a - 1)^{a|V(G)|}\). Given any graph G and \(a \in {\mathbb {N}}\), we wish to determine the smallest b such that \(\chi _\ell (G \square K_{a,b}) = \chi _\ell (G) + a\). In this paper we show that the list color function, a list analogue of the chromatic polynomial, provides the right concept and tool for making progress on this problem. Using the list color function, we prove a general improvement on Borowiecki et al.’s (2006) result, and we compute the smallest such b for some large families of chromatic-choosable graphs.