The $b$-Chromatic Number and $f$-Chromatic Vertex Number of Regular Graphs

TL;DR

This paper investigates the $b$-chromatic number and $f$-chromatic vertex number in regular graphs, exploring a conjecture relating girth and chromatic properties, and provides partial results under additional conditions.

Contribution

It offers new insights and partial answers to the conjecture that $b(G)=d+1$ for $d$-regular graphs of girth 5, under supplementary conditions.

Findings

Partial confirmation of the conjecture under specific conditions.

New bounds for the $b$-chromatic number in regular graphs.

Insights into the structure of dominant vertices in regular graphs.

Abstract

The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the largest positive integer $k$ such that there exists a proper coloring for G with $k$ colors in which every color class contains at least one vertex adjacent to some vertex in each of the other color classes, such a vertex is called a dominant vertex. The $f$-chromatic vertex number of a $d$-regular graph $G$, denoted by $f(G)$, is the maximum number of dominant vertices of distinct colors in a proper coloring with $d+1$ colors. El Sahili and Kouider conjectured that $b(G)=d+1$ for any $d$-regular graph $G$ of girth 5. We study this conjecture by giving some partial answers under supplementary conditions.