An Upper Bound of the Adjacent-vertex-strong-distinguishing Total Chromatic Number of Graphs

Abstract

A proper total coloring of the graph G is called adjacent-vertex-strong-distin- guishing total coloring, if any two adjacent vertices have different color sets, where the color set of a vertex u is the set composed of all colors of u and the edges and vertices incident to u. On the base of the bound of adjacent-vertex-strong-distinguishing total chromatic number (�ast(G) � 2�(G)+1), the new upper bounds of the adjacent-vertex-strong-distinguishing total chromatic number of the graph G is obtained by the way of probability, where G is a simple graph with no isolated edge and �(G) � 3.