List-Coloring Squares of Sparse Subcubic Graphs

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 $g\ge 24$, at most 5 if $g\ge 14$, and at most 6 if $g\ge 10$. 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.