The adjacent vertex distinguishing total chromatic numbers of planar graphs with $\Delta =10$

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 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. A total-k-adjacent vertex distinguishing-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 $\chi ''_{a}(G)$. It is known that $\chi _{a}''(G)\le \Delta (G)+3$ for any planar graph with $\Delta (G)\ge 11$. In this paper, we show that if G is a planar graph with $\Delta (G)\ge 10$, then $\chi _{a}''(G)\le \Delta (G)+3$. Our approach is based on Combinatorial Nullstellensatz and the discharging method.

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