On the b-chromatic number of star graph operators

TL;DR

This paper investigates the b-chromatic number of the power, total, and line graphs derived from the Cartesian product of star graphs, providing exact values and bounds for these parameters.

Contribution

It generalizes previous results on the b-chromatic number of Cartesian products of star graphs and explores new bounds for related graph products.

Findings

Exact b-chromatic numbers for specific graph products

Generalized bounds for Cartesian products of complete graphs

Extension of previous results to broader graph classes

Abstract

A $b$-coloring is a proper coloring such that for each color class, there exists at least one vertex that is adjacent to at least one vertex in every other color class. The $b$-chromatic number of a graph $G$ is the maximum number $k$ such that $G$ admits a $b$-coloring with $k$ colors. This paper focuses on the $b$-chromatic number of the power graph of the Cartesian product of star graphs. In addition, we also study the total graph and the line graph of the Cartesian product of star graphs. Our main result generalizes the result shown in \cite{qn} on the b-chromatic number of the Cartesian product of two stars. We find exact values for the b-chromatic number of particular Cartesian products of complete graphs and explore the bounds of the generalized Cartesian product of complete graphs.