The DP-coloring of the square of subcubic graphs

TL;DR

This paper investigates the DP-coloring of the square of subcubic graphs, establishing bounds based on maximum average degree and demonstrating the sharpness of these bounds.

Contribution

It provides new bounds for DP-coloring of the square of subcubic graphs using discharging, extending understanding of coloring under degree constraints.

Findings

If mad(G)<9/4, then G^2 is DP-5-colorable.

If mad(G)<12/5, then G^2 is DP-6-colorable.

The bound for DP-5-colorability is proven to be sharp.

Abstract

The 2-distance coloring of a graph $G$ is equivalent to the proper coloring of its square graph $G^2$, it is a special distance labeling problem. DP-coloring (or "Correspondence coloring") was introduced by Dvo\v{r}\'ak and Postle in 2018, to answer a conjecture of list coloring proposed by Borodin. In recent years, many researches pay attention to the DP-coloring of planar graphs with some restriction in cycles. We study the DP-coloring of the square of subcubic graphs in terms of maximum average degree $\rm{mad}(G)$, and by the discharging method, we showed that: for a subcubic graph $G$, if $\rm{mad}(G)<9/4$, then $G^2$ is DP-5-colorable; if $\rm{mad}(G)<12/5$, then $G^2$ is DP-6-colorable. And the bound in the first result is sharp.