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

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