What question did this study set out to answer?

The aim is to explore total dominator coloring in the lexicographic product of graphs and establish important bounds and relationships.

March 5, 2026Open Access

Total dominator coloring of the lexicographic product of graphs

Key Points

The aim is to explore total dominator coloring in the lexicographic product of graphs and establish important bounds and relationships.
Identified graphs with equal total dominator chromatic numbers.
Showed cases where the total dominator chromatic number equals standard chromatic numbers.
Presented four upper bounds and a lower bound for the total dominator chromatic number.
Developed a sufficient condition for chromatic number equality in specific graph products.
Established a Nordhaus-Gaddum-like relation for the total dominator chromatic number.
Found examples of graphs with identical total dominator chromatic numbers.
Demonstrated relationships between total dominator chromatic numbers and chromatic numbers.
Established four upper bounds and one lower bound for specific graph products.
Described conditions under which the graph's total dominator chromatic number equals that of its product with a complete graph.
Calculated total dominator chromatic numbers for products involving stars, wheels, and complete graphs.

Abstract

A total dominator coloring of a graph which has no isolated vertex is a proper vertex coloring of the graph in which each vertex of the graph is adjacent to all vertices of some (other) color class. The total dominator chromatic number of a graph is the minimum size of color classes in a total dominator coloring of the graph. This paper establishes the total dominator coloring of the lexicographic product of graphs. Firstly, we present some graphs whose total dominator chromatic numbers are same. Also we show some graphs whose total dominator chromatic numbers are equal to their chromatic numbers. Then, after presenting four upper bounds and a lower bound for the total dominator chromatic number of the lexicographic product of two graphs, we give a sufficient condition for that the total dominator chromatic number of the lexicographic product of two graphs to be equal to the total dominator chromatic number of the lexicographic product of the first graph to a complete graph. We next establish a Nordhaus-Gaddum-like relation for the total dominator chromatic number of the lexicographic product of two graphs, and finally find the total dominator chromatic number of the lexicographic product of a star, a wheel or a complete graph with an arbitrary graph.

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