List-coloring the Square of a Subcubic Graph

Abstract

The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set as $G$ and with two vertices adjacent if their distance in $G$ is at most 2. Thomassen showed that every planar graph $G$ with maximum degree $\Delta(G)=3$ satisfies $\chi(G^2)\leq 7$. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of $G^2$ equals the chromatic number of $G^2$, that is $\chi_l(G^2)=\chi(G^2)$ for all $G$. If true, this conjecture (together with Thomassen's result) implies that every planar graph $G$ with $\Delta(G)=3$ satisfies $\chi_l(G^2)\leq 7$. We prove that every connected graph (not necessarily planar) with $\Delta(G)=3$ other than the Petersen graph satisfies $\chi_l(G^2)\leq 8$ (and this is best possible). In addition, we show that if $G$ is a planar graph with $\Delta(G)=3$ and girth $g(G)\geq 7$, then $\chi_l(G^2)\leq 7$. Dvo\v{r}\'ak, \v{S}krekovski, and Tancer showed that if $G$ is a planar graph with $\Delta(G) = 3$ and girth $g(G) \geq 10$, then $\chi_l(G^2)\leq 6$. We improve the girth bound to show that if $G$ is a planar graph with $\Delta(G)=3$ and $g(G) \geq 9$, then $\chi_l(G^2) \leq 6$. All of our proofs can be easily translated into linear-time coloring algorithms.