Constructing Regular Graphs with Smallest Defining Number

Abstract

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 cardinality 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 $n\ge 3k$, if $r\ge 2(k-1)$ then $d(n,r,\chi =k)=k-1$. In this paper we show that for a given $k$ and for all $n<3k$ and $r\ge 2(k–1)$, $d(n,r,\chi =k)=k-1$.