More on the total dominator chromatic number of a graph

Abstract

Let G be a simple graph. A total dominator coloring of G, is a proper coloring of the vertices of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number of G, is the minimum number of colors among all total dominator coloring of G. The neighbourhood corona of two graphs G1 and G2 is denoted by G1 ⋆ G2 and is the graph obtained by taking one copy of G1 and |V(G1)| copies of G2, and joining the neighbours of the ith vertex of G1 to every vertex in the ith copy of G2. In this paper, we study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of r-gluing of two graphs. Stability (bondage number) of total dominator chromatic number of G is the minimum number of vertices (edges) of G whose removal changes the TDC-number of G. We study the stability and bondage number of certatin graphs.