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List-coloring the square of a subcubic graph. (English) Zbl 1172.05023
Summary: The square $$G^2$$ of a graph $$G$$ is the graph with the same vertex set $$G$$ 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 $$\chi(G^2)\leq7$$. 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 $$\chi_l(G^2)\leq7$$. 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 $$g(G)\geq7$$, then $$\chi_l(G^2)\leq7$$. Dvořák, Škrekovski, and Tancer showed that if $$G$$ is a planar graph with $$\Delta(G)=3$$ and girth $$g(G)\geq10$$, then $$\chi_l(G^2)\geq 6$$. 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.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C12 Distance in graphs
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