On Adjacent Vertex-distinguishing Total Chromatic Number of Generalized Mycielski Graphs

The adjacent vertex-distinguishing total chromatic number of a graph G, denoted by Eat(G), is the smallest k for which G has a proper total k-coloring such that any two adjacent vertices have distinct sets of colors appearing on the vertex and its incident edges. In regard of this number, there is a famous conjecture (AVDTCC) which states that for any simple graph G, (Xat)(G)<=Delta(G)+3. In this paper, we study this number for the generalized Mycielski graph (mu m)(G) of a graph G. We prove that the satisfiability of the conjecture AVDTCC in G implies its satisfiability in (mu m)(G). Particularly we give the exact values of Eat((mu m)(G)) when G is a graph with maximum degree less than 3 or a complete graph. Moreover, we investigate Eat(G) for any graph G with only one maximum degree vertex by showing that Eat(G)<=Delta(G) + 2 when Delta(G)<= 4.