Total colorings of certain classes of lexicographic product graphs

Abstract

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, [Formula: see text] is the minimum number of colors which need to be assigned to obtain a total coloring of the graph [Formula: see text]. The Total Coloring Conjecture made independently by Behzad and Vizing claims that, [Formula: see text], where [Formula: see text] represents the maximum degree of [Formula: see text]. The lower bound is sharp, the upper bound remains to be proved. In this paper, we prove the Total Coloring Conjecture for certain classes of lexicographic product and deleted lexicographic product of graphs.

Total coloring conjecture on certain classes of product graphs

“…Direct product, cartesian product, strong product and lexicographic product graphs given by Imrich [8] et la. Recently, Vignesh et al [21,16] verified TCC for certain classes of deleted lexicogaphic product graphs. In [20], they also proved that Vertex, Edge and Neighborhood corona products of graphs are type-I graphs.…”