Sufficient sparseness conditions for G^2 to be (Δ+1)-choosable, when Δ\ge5

Abstract

We determine the list chromatic number of the square of a graph $\chil(G^2)$ in terms of its maximum degree $\Delta$ when its maximum average degree, denoted $\mad(G)$, is sufficiently small. For $\Delta \ge 6$, if $\mad(G)<2+\frac{4Δ-8}{5Δ+2}$, then $\chil(G^2)=Δ+1$. In particular, if $G$ is planar with girth $g\ge 7+\frac{12}{\Delta -2}$, then $\chil(G^2)=Δ+1$. Under the same conditions, $\chil^i(G)=Δ$, where $\chil^i$ is the list injective chromatic number.

arXiv comment16 pages, 6 figures; incorporated reviewer feedback