Adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least 10

Abstract

A (proper) total-k-coloring $$\phi :V(G)\cup E(G)\rightarrow \{1,2,\ldots ,k\}$$ is called adjacent vertex distinguishing if $$C_{\phi }(u)\ne C_{\phi }(v)$$ for each edge $$uv\in E(G)$$ , where $$C_{\phi }(u)$$ is the set of the color of u and the colors of all edges incident with u. We use $$\chi ''_a(G)$$ to denote the smallest value k in such a coloring of G. Zhang et al. first introduced this coloring and conjectured that $$\chi ''_a(G)\le \Delta (G)+3$$ for any simple graph G. For the list version of this coloring, it is known that $$ch''_a(G)\le \Delta (G)+3$$ for any planar graph with $$\Delta (G)\ge 11$$ , where $$ch''_a(G)$$ is the adjacent vertex distinguishing total choosability. In this paper, we show that if G is a planar graph with $$\Delta (G)\ge 10$$ , then $$ch''_a(G)\le \Delta (G)+3$$ .