List coloring the square of sparse graphs with large degree

TL;DR

This paper extends coloring results for the squares of sparse graphs with large maximum degree, establishing a new bound based on maximum average degree less than 3, which is shown to be optimal.

Contribution

It proves that for graphs with maximum average degree less than 3, the square is list $( riangle+1)$-colorable if the maximum degree is sufficiently large, strengthening previous planar graph results.

Findings

Bound of 3 on maximum average degree is optimal.

Squares of such graphs are list $( riangle+1)$-colorable for large enough degree.

Results apply to list injective $ riangle$-coloring as well.

Abstract

We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree $\Delta$ are list $(\Delta+1)$-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Euler's Formula, planar graphs of girth at least $6$ are of maximum average degree less than $3$, and planar graphs of girth at least $7$ are of maximum average degree less than $14/5<3$. We strengthen their result and prove that there exists a function $f$ such that the square of any graph with maximum average degree $m<3$ and maximum degree $\Delta\geq f(m)$ is list $(\Delta+1)$-colorable. This bound of $3$ is optimal in the sense that the above-mentioned planar graphs with girth $6$ have maximum average degree less than $3$ and arbitrarily large maximum degree, while their square cannot be $(\Delta+1)$-colored. The same holds for list injective $\Delta$-coloring.