On The b-Chromatic Number of Regular Bounded Graphs

TL;DR

This paper proves a conjecture about the b-chromatic number of certain regular graphs, showing it equals d+1 for large enough graphs with specific girth and cycle restrictions, thus advancing understanding of graph coloring properties.

Contribution

It confirms the conjecture that the b-chromatic number equals d+1 for large d-regular graphs with girth 5 and no 4-cycles, improving previous bounds.

Findings

b(G)=d+1 for graphs with at least d^3+d vertices and no 4-cycle

Confirmed the conjecture for graphs with girth 5 and sufficient size

Improved bounds on the number of vertices needed for the conjecture to hold

Abstract

A $b$-coloring of a graph is a proper coloring such that every color class contains a vertex adjacent to at least one vertex in each of the other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the maximum integer $k$ such that $G$ admits a $b$-coloring with $k$ colors. El Sahili and Kouider conjectured that $b(G)=d+1$ for $d$-regular graph with girth 5, $d\geq4$. In this paper, we prove that this conjecture holds for $d$-regular graph with at least $d^3+d$ vertices. More precisely we show that $b(G)=d+$1 for $d$-regular graph with at least $d^3+d$ vertices and containing no cycle of order 4. We also prove that $b(G)=d+1$ for $d$-regular graphs with at least $2d^3+2d-2d^2$ vertices improving Cabello and Jakovac bound.