A note on the adjacent vertex distinguishing total chromatic number of graphs

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 different sets of colors. The minimum number of colors needed for such a total coloring of G is denoted by χ a ″ ( G ) . In this note, we show that χ a ″ ( G ) ≤ 2 Δ for any graph G with maximum degree Δ ≥ 3 .