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- W2031929504 abstract "The problem of coloring the square of a graph naturally arises in connection with the distance labelings, which have been studied intensively. We consider this problem for sparse subcubic graphs. We show that the choosability $chi_ell(G^2)$ of the square of a subcubic graph G of maximum average degree d is at most four if $d<24/11$ and G does not contain a 5-cycle, at most five if $d<7/3$, and at most six if $d<5/2$. Wegner's conjecture claims that the chromatic number of the square of a subcubic planar graph is at most seven. Let G be a planar subcubic graph of girth g. Our result implies that $chi_ell(G^2)$ is at most four if $gge 24$, at most 5 if $gge 14$, and at most 6 if $gge 10$. For lower bounds, we find a planar subcubic graph $G_1$ of girth 9 such that $chi(G_1^2)=5$ and a planar subcubic graph $G_2$ of girth 5 such that $chi(G_2^2)=6$. As a consequence, we show that the problem of 4-coloring of the square of a subcubic planar graph of girth $g=9$ is NP-complete. We conclude the paper by posing a few conjectures." @default.
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- W2031929504 date "2008-01-01" @default.
- W2031929504 modified "2023-10-10" @default.
- W2031929504 title "List-Coloring Squares of Sparse Subcubic Graphs" @default.
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- W2031929504 doi "https://doi.org/10.1137/050634049" @default.
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