A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements receive the same color. The total chromatic number of a graph G, denoted by , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G, , where is the maximum degree of G. In this paper, we prove the total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.
R Vignesh, Geetha, J., and Dr. Somasundaram K., “Total Coloring Conjecture for Certain Classes of Graphs”, Algorithms, vol. 11, p. 161, 2018.