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 $ch''_a(G)\le \Delta (G)+3$.

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