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 χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by d(G, χ). Let d(n, r, χ = 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, χ = 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, χ = k) = k − 1.