Dominator Coloring of Total Graph of Path and Cycle

A dominator coloring of a graph G is a proper coloring in which every vertex of G dominates every vertex of at least one-color class possibly its own class and each color class is dominated by at least one vertex. The minimum number of colors required for dominator coloring of G is called the dominator chromatic number of G and is denoted by χdG. In this paper, we have established the relation between dominator chromatic number χdG, chromatic number χG and domination number γG. We have investigated results on total graphs of path and cycle with χdG = χ G + γ G and χdG = χ G + γ G - 1.