The adjacent vertex distinguishing total coloring of planar graphs

Abstract

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by χa′(G). In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ≥14, then Δχa(G)Δ+2, and χa (G)=Δ+2 if and only if G contains two adjacent vertices of maximum degree. © 2012 Springer Science+Business Media, LLC.