Criticality, The List Color Function, and List Coloring the Cartesian Product of Graphs

We introduce a notion of color-criticality in the context of chromatic-choosability. We define a graph $G$ to be strong $k$-chromatic-choosable if $\chi(G) = k$ and every $(k-1)$-assignment for which $G$ is not list-colorable has the property that the lists are the same for all vertices. That is the usual coloring is, in some sense, the obstacle to list-coloring. We prove basic properties of strongly chromatic-choosable graphs such as chromatic-choosability and vertex-criticality, and we construct infinite families of strongly chromatic-choosable graphs. We derive a sufficient condition for the existence of at least two list colorings of strongly chromatic-choosable graphs and use it to show that: if $M$ is a strong $k$-chromatic-choosable graph with $|E(M)| \leq |V(M)|(k-2)$ and $H$ is a graph that contains a Hamilton path, $w_1, w_2, \ldots, w_m$, such that $w_i$ has at most $\rho \geq 1$ neighbors among $w_1, \ldots, w_{i-1}$, then $\chi_{\ell}(M \square H) \le k+ \rho - 1$. We show that this bound is sharp for all $\rho \ge 1$ by generalizing the theorem to apply to $H$ that are $(M,\rho)$-Cartesian accommodating which is a notion we define with the help of the list color function, $P_{\ell}(G,k)$, the list analogue of the chromatic polynomial. We also use the list color function to determine the list chromatic number of certain star-like graphs: $\chi_{\ell}(M \square K_{1,s}) =$ $k \; \text{if } s < P_{\ell}(M,k)$, or $k+1 \; \text{if } s \geq P_{\ell}(M,k)$, where $M$ is a strong $k$-chromatic-choosable graph. We show that $P_{\ell}(M,k)$ equals $P(M,k)$, the chromatic polynomial, when $M$ is an odd cycle, complete graph, or the join of an odd cycle with a complete graph.