Weak degeneracy of the square of K4-minor free graphs

Abstract

A graph G is called weakly f-degenerate with respect to a function f from $V\left(G\right)$ to the non-negative integers, if every vertex of G can be successively removed through a series of valid Delete and DeleteSave operations. The weak degeneracy $wd\left(G\right)$ is defined as the smallest integer d for which G is weakly d-degenerate, where d is a constant function. It was demonstrated that one plus the weak degeneracy can act as an upper bound for list-chromatic number and DP-chromatic number. Let $\kappa \left(G^2\right)=\Delta \left(G\right)+2$ if $2\le \Delta \left(G\right)\le 3$, and $\kappa \left(G^2\right)=\lfloor \frac{3\Delta \left(G\right)}{2}\rfloor$ if $\Delta \left(G\right)\ge 4$. In this paper, we prove that for every $K_4$-minor free graph G, $wd\left(G^2\right)\le \kappa \left(G^2\right)$, which implies that $G^2$ is $\left(\kappa \right(G^2)+1)$-choosable and $\left(\kappa \right(G^2)+1)$-DP-colorable. This work generalizes the result obtained by Lih et al. in [Discrete Mathematics, 269 (2003), 303-309] and Hetherington et al. in [Discrete Mathematics, 308 (2008), 4037-4043].