Adjacent vertex distinguishing total colorings of outer 1-planar graphs

摘要

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that no pair of adjacent vertices are incident to the same set of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by X~＂_a(G). In this paper, we prove that if G is an outer 1-planar graph with at least two vertices, then X~＂_a(G) < max{A + 2,8}. Moreover, we also prove that when A > 7, x~＂_a(G) = A + 2 if and only if G contains two adjacent vertices of maximum degree.