Total chromatic number for certain classes of product graphs

Total coloring is a function that assigns colors to the vertices and edges of the graph such that the adjacent and the incident elements receive different colors. The minimum number of colors required for a proper total coloring of a graph G is called the total chromatic number of G and is denoted by chi ''(G). Behzad-Vizing conjecture (Total Coloring Conjecture) states that for any graph G, chi ''(G) <= Delta(G) + 2, where Delta(G) is the maximum degree of G. In this paper, we verify the Behzad-Vizing conjecture for some product graphs.