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 $\mathrm{\mid }E(M)\mathrm{\mid }\le \mathrm{\mid }V(M)\mathrm{\mid }(k-2)$ and $H$ is a graph that contains a Hamilton path, $w_1,w_2,\dots ,w_m$, such that $w_i$ has at most $\rho \ge 1$ neighbors among $w_1,\dots ,w_{i-1}$, then $\chi_{\ell}(M \square H) \le k+ \rho - 1$\mathrm{.}Weshowthatthisboundissharpforall$\rho \ge 1$bygeneralizingthetheoremtoapplyto$H$thatare$(M,\rho)$-Cartesian accommodating which is a notion we define with the help of the list color function, $P_{\mathrm{\ell }}(G,k)$, 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: ${\chi }_{\mathrm{\ell }}(M\mathrm{\square }{K}_{1,s})=$ $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 $P_{\mathrm{\ell }}(M,k)$ 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.