On the b-chromatic number of the Cartesian product of two complete graphs

TL;DR

This paper investigates the b-chromatic number of the Cartesian product of two complete graphs, providing counterexamples to a previous conjecture for specific values of n.

Contribution

It disproves the conjecture that the b-chromatic number is always 2n-3 for all n ≥ 5 by presenting counterexamples for n=5, 6, and 7.

Findings

Counterexamples for n=5, 6, 7

Disproof of the conjecture for these cases

Insights into the b-chromatic number behavior

Abstract

A b-coloring of a graph $G$ is a coloring of its vertices such that every color class contains a vertex that has neighbors in all other classes. The b-chromatic number of $G$ is the largest integer $k$ such that $G$ has a b-coloring with $k$ colors. Javadi and Omoomi ("On b-coloring of cartesian product of graphs", Ars Combinatoria 107 (2012) 521-536) proved that the b-chromatic number of $K_n \times K_n$ (the Cartesian product of two complete graphs on $n$ vertices) is in the set $\{2n-3, 2n-2\}$ and conjectured that the exact value is $2n-3$ for all $n \ge 5$. We give counterexamples to this conjecture for $n=5$, $n=6$ and $n=7$.