Toat Dominator Chromatic number of a Graph

TL;DR

This paper investigates the total dominator chromatic number of graphs, providing bounds, characterizations, and exploring its relation to other graph invariants across various graph classes.

Contribution

It introduces the total dominator coloring problem, analyzes its properties, and establishes bounds and relationships with chromatic and total domination numbers.

Findings

Established bounds for total dominator chromatic number

Characterized this number for specific graph classes

Linked total dominator chromatic number to chromatic and total domination numbers

Abstract

Given a graph $G$, the total dominator coloring problem seeks a proper coloring of $G$ with the additional property that every vertex in the graph is adjacent to all vertices of a color class. We seek to minimize the number of color classes. We study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also show the relation between total dominator chromatic number and chromatic number and total domination number.