The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven

Abstract

A (proper) total-k-coloring of a graph G is a mapping $\phi : V (G) \cup E(G)\mapsto \{1, 2, \ldots , k\}$ such that any two adjacent or incident elements in $V (G) \cup E(G)$ receive different colors. Let C(v) denote the set of the color of a vertex v and the colors of all incident edges of v. An adjacent vertex distinguishing total-k-coloring of G is a total-k-coloring of G such that for each edge $uv\in E(G)$, $C(u)\ne C(v)$. We denote the smallest value k in such a coloring of G by $\chi ^{\prime \prime }_{a}(G)$. It is known that $\chi _{a}^{\prime \prime }(G)\le \Delta (G)+3$ for any planar graph with $\Delta (G)\ge 10$. In this paper, we consider the list version of this coloring and show that if G is a planar graph with $\Delta (G)\ge 11$, then ${ ch}_{a}^{\prime \prime }(G)\le \Delta (G)+3$, where ${ ch}^{\prime \prime }_a(G)$ is the adjacent vertex distinguishing total choosability.