List Coloring a Cartesian Product with a Complete Bipartite Factor

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 $b \geq a^a$. Since $\chi_\ell(K_{a,b}) \leq 1 + a$ 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 $\chi_\ell(G \square K_{a,b}) \leq \chi_\ell (G) + a$. In 2006, Borowiecki, Jendrol, Kr\'al, and Miskuf showed that this bound is attainable if $b$ is sufficiently large; specifically, $\chi_\ell(G \square K_{a,b}) = \chi_\ell (G) + a$ whenever $b \geq (\chi_\ell(G) + a - 1)^{a|V(G)|}$. 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 $b$ exactly for some large families of chromatic-choosable graphs.