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In a given graph $G$, a set $S$ of vertices with an assignment of colors is a{\sf defining set of the vertex coloring of $G$}, if there exists a uniqueextension of the colors of $S$ to a $\Cchi(G)$-coloring of the vertices of $G$.A defining set with minimum cardinality is called a {\sf smallest defining set}(of vertex coloring) and its cardinality, the {\sf defining number}, is denotedby $d(G, \Cchi)$. Let $ d(n, r, \Cchi = k)$ be the smallest defining number ofall $r$-regular $k$-chromatic graphs with $n$ vertices. Mahmoodian et. al\cite{rkgraph} proved that, for a given $k$ and for all $n \geq 3k$, if $r \geq2(k-1)$ then $d(n, r, \Cchi = k)=k-1$. In this paper we show that for a given$k$ and for all $n < 3k$ and $r\geq 2(k-1)$, $d(n, r, \Cchi=k)=k-1$.

Constructing regular graphs with smallest defining number