The adjacent vertex distinguishing total chromatic numbers of planar graphs with Δ=10

A (proper) total‐k‐coloring of a graph G is a mapping $\mathit{\varphi }:V(G)\cup E(G)\mapsto \{1,2,\dots ,k\}$ 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 ${\mathit{\chi }}_a^{\prime \prime }(G)$ . It is known that ${\mathit{\chi }}_a^{\prime \prime }(G)\le \mathrm{\Delta }(G)+3$ for any planar graph with $\mathrm{\Delta }(G)\ge 11$ . In this paper, we show that if G is a planar graph with $\mathrm{\Delta }(G)\ge 10$ , then ${\mathit{\chi }}_a^{\prime \prime }(G)\le \mathrm{\Delta }(G)+3$ . Our approach is based on Combinatorial Nullstellensatz and the discharging method.