Total Chromatic Number for Certain Classes of Lexicographic Product Graphs

A total coloring of a graph G is an assignment of colors to all the elements (vertices and edges) of the graph in such a way that no two adjacent or incident elements receive the same color. The total chromatic number of G, denoted by χ (G), is the minimum number of colors which need for total coloring of G. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing which claims that, Δ(G)+1 ≤ χ ≤ Δ(G)+2 , where Δ(G) is the maximum degree of G. The lower bound is sharp and the upper bound remains to be proved. In this paper, we prove the TCC for certain classes of lexicographic and deleted lexicographic products of graphs. Also, we obtained the lower bound for certain classes of these products.