Criticality, the list color function, and list coloring the cartesian product of graphs

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