Two-distance vertex-distinguishing total coloring of subcubic graphs

Abstract

A 2-distance vertex-distinguishing total coloring of graph G is a proper total coloring of G such that any pair of vertices at distance of two have distinct sets of colors. The 2-distance vertex-distinguishing total chromatic number $\chi_{d2}^{''}(G)$ of G is the minimum number of colors needed for a 2-distance vertex-distinguishing total coloring of G. In this paper, it's proved that if G is a subcubic graph, then $\chi_{d2}^{''}(G)\le 7$.