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 ${\chi }_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1\star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. 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.

arXiv comment11 pages, 8 pictures