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

TL;DR

This paper establishes conditions based on maximum average degree and girth for the square of a graph with maximum degree  to be (+1)-choosable, extending understanding of graph coloring in sparse graphs.

Contribution

It provides new sufficient conditions involving maximum average degree and girth for the list chromatic number of graph squares to equal +1, especially for planar graphs.

Findings

(G^2)=+1 under certain  degree and sparsity conditions

Planar graphs with large girth satisfy (G^2)=+1

List injective chromatic number equals  under the same conditions

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 $\mad(G)<2+\frac{4\Delta-8}{5\Delta+2}$, then $\chil(G^2)=\Delta+1$. In particular, if $G$ is planar with girth $g\ge 7+\frac{12}{\Delta-2}$, then $\chil(G^2)=\Delta+1$. Under the same conditions, $\chil^i(G)=\Delta$, where $\chil^i$ is the list injective chromatic number.