On Total Coloring of Triple Star and Lobster Graphs

Abstract

A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G such that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which G has a k-total coloring. The well-known Total Coloring Conjecture asserts that the total chromatic number of a graph is either ∆(G) + 1 or ∆(G) + 2, where ∆(G) is the maximum degree of G. In this paper, we consider the triple star graph, lobster graph and its line, middle, total graphs and also splitting graph of triple star.  We obtained the preceding graphs has total chromatic number equal to ∆(G) + 1.