Constructing regular graphs with smallest defining number

In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a chi(G)-coloring of the vertices of G. A defining set with minimum caxdinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by d(G, chi). Let d(n,r, chi = k) be the smallest defining number of all r-regular k-chromatic graphs with n vertices. Mahmoodian et. al [7] proved that, for a given k and for all n >= 3k, if r >= 2(k - 1) then d(n, r, chi = k) = k - 1. In this paper we show that for a given k and for all n < 3k and r >= 2(k - 1), d(n,r, chi = k) = k - 1.