  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.
