Concise proofs for adjacent vertex-distinguishing total colorings

Let G = ( V , E ) be a graph and f : ( V ∪ E ) → [ 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 χ a t ( 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 χ a t ( G ) when G is a complete graph or a cycle.