The $b$-chromatic number and $f$-chromatic vertex number of regular graphs

Abstract

The $b$-chromatic number of a graph $G$, denoted by $b\left(G\right)$, 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 $f\left(G\right)$, is the maximum number of dominant vertices of distinct colors in a proper coloring with $d+1$ colors. El Sahili and Kouider conjectured that $b\left(G\right)=d+1$ for any $d$-regular graph $G$ of girth 5. Blidia, Maffray and Zemir (2009) reformulated this conjecture by excluding the Petersen graph and proved it for $d\le 6$. We study El Sahili and Kouider conjecture by giving some partial answers under supplementary conditions.