Total dominator coloring of central graphs

TL;DR

This paper investigates the total dominator coloring of central graphs, establishing bounds, relations, and exact values for various graph classes to advance understanding of this coloring concept.

Contribution

It provides new bounds, relations, and exact calculations for the total dominator chromatic number of central graphs and their specific instances.

Findings

Established tight bounds for the total dominator chromatic number of central graphs.

Derived Nordhaus-Gaddum-like relations for the total dominator chromatic number.

Calculated exact values for specific graph classes such as paths, cycles, and complete graphs.

Abstract

A total dominator coloring of a graph $G$ is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring of it. Here, we study the total dominator coloring on central graphs by giving some tight bounds for the total dominator chromatic number of the central of a graph, join of two graphs and Nordhaus-Gaddum-like relations. Also we will calculate the total dominator chromatic number of the central of a path, a cycle, a wheel, a complete graph and a complete multipartite graph.