On the adjacent vertex distinguishing total coloring numbers of graphs with Δ=3Δ=3

An adjacent vertex distinguishing total-coloring of a simple graph GG is a proper total-coloring of GG such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χa″(G) required to give GG an adjacent vertex distinguishing total-coloring is studied. We proved χa″(G)⩽6 for graphs with maximum degree Δ(G)=3Δ(G)=3 in this paper.