List-coloring the square of a subcubic graph.

*(English)*Zbl 1172.05023Summary: 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.

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