The DP-coloring of the square of subcubic graphs

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 Dvořák 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 $\mathrm{m}\mathrm{a}\mathrm{d}(G)$, and by the discharging method, we showed that: for a subcubic graph $G$, if $\mathrm{m}\mathrm{a}\mathrm{d}(G)<9\mathrm{/}4$, then $G^2$ is DP-5-colorable; if $\mathrm{m}\mathrm{a}\mathrm{d}(G)<12\mathrm{/}5$, then $G^2$ is DP-6-colorable. And the bound in the first result is sharp.