A Note on the Adjacent Vertex Distinguishing Total Chromatic Number of Graph
A total coloring of a simple graph G is called adjacent vertex distinguishing if for any two adjacent and distinct vertices u and v in G, the set of colors assigned to the vertices and the edges incident to u differs from the set of colors assigned to the vertices and the edges incident to v. In this paper we shall prove the series-parallel graph with maximum degree 3 and the series-parallel graph whose the number of edges is the double of maximum degree minus 1 satisfy the adjacent vertex distinguishing total coloring conjecture.
Z. W. Wang "A Note on the Adjacent Vertex Distinguishing Total Chromatic Number of Graph", Key Engineering Materials, Vols. 474-476, pp. 2341-2345, 2011