Bounds for the b-chromatic number of powers of hypercubes

TL;DR

This paper investigates the b-chromatic number of powers of hypercubes, providing improved upper bounds and establishing lower bounds for related Hamming graphs, advancing understanding of graph coloring properties.

Contribution

It offers new upper bounds for the b-chromatic number of powers of hypercubes and determines lower bounds for the b-chromatic number of certain Hamming graphs, addressing open problems.

Findings

Improved upper bounds for powers of hypercubes.

Lower bounds for b-chromatic number of Hamming graphs.

Advances in graph coloring theory for hypercubes and Hamming graphs.

Abstract

The b-chromatic number $b(G)$ of a graph $G$ is the maximum $k$ for which $G$ has a proper vertex coloring using $k$ colors such that each color class contains at least one vertex adjacent to a vertex of every other color class. In this paper, we mainly investigate on one of the open problems given in [P. Francis, S. Francis Raj, On b-coloring of powers of hypercubes, Discrete Appl. Math. 225 (2017) 74-86.]. As a consequence, we have obtained an upper bound for the b-chromatic number of some powers of hypercubes. This turns out to be an improvement of the already existing bound in [P. Francis, S. Francis Raj, On b-coloring of powers of hypercubes, Discrete Appl. Math. 225 (2017) 74-86.]. Further, we have determined a lower bound for the b-chromatic number of some powers of the Hamming graph, a generalization of the hypercube.