Concise proofs for adjacent vertex-distinguishing total colorings

Let G = (V, E) be a graph and f : (VUE) -> [k] be a proper total k-coloring of G. We say that f is an adjacent vertex- distinguishing total coloring if for any two adjacent vertices, the set of colors appearing on the vertex and incident edges are different. We call the smallest k for which such a coloring of G exists the adjacent vertex-distinguishing total chromatic number, and denote it by chi(at)(G). Here we provide short proofs for an upper bound on the adjacent vertex-distinguishing total chromatic number of graphs of maximum degree three, and the exact values of chi(at)(G) when G is a complete graph or a cycle. (c) 2008 Elsevier B.V. All rights reserved.