Smallest Defining Number of \(r\)-Regular \(k\)-Chromatic Graphs: \(r \neq k\)

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 and Mendelsohn (1999) proved that for each \(n\geq m\) and each \(r \geq 4\), \(d(n,r, \chi = 3) = 2\). They raised the following question: Is it true that for every \(k\), there exist \(n_0(k)\) and \(r_0(k)\), such that for all \(n \geq n_0(k)\) and \(r \geq r_0(k)\) we have \(d(n,r, \chi = k) = k-1\)? We show that the answer to this question is positive, and we prove that for a given \(k\) and for all \(n \geq 3k\), if \(r \geq 2(k – 1)\) then \(d(n,r, \chi = k) = k-1\).