Total dominator chromatic number of specific graphs

TL;DR

This paper investigates the total dominator chromatic number, a graph coloring parameter where each vertex is adjacent to all vertices of some color class, focusing on specific graph families.

Contribution

It introduces the concept of total dominator chromatic number and analyzes its values for particular classes of graphs, expanding understanding of graph coloring constraints.

Findings

Determined $ ext{chi}_d^t(G)$ for certain graph classes.

Established bounds for the total dominator chromatic number.

Provided exact values for specific graphs.

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 number $\chi_d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. In this paper, we study the total dominator chromatic number of some specific graphs.