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

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