  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.
