Constructing regular graphs with smallest defining number

TL;DR

This paper investigates the smallest defining number for vertex colorings in regular graphs, establishing new bounds for graphs with fewer than 3k vertices and degree at least 2(k-1).

Contribution

It extends previous results by determining the minimal defining number for a broader class of regular graphs with fewer vertices.

Findings

For all n < 3k and r ≥ 2(k-1), the smallest defining number is k-1.

Generalizes earlier bounds to include graphs with fewer than 3k vertices.

Provides a complete characterization of the minimal defining number in the specified parameter range.

Abstract

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 unique extension 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 denoted by $d(G, \Cchi)$. Let $ d(n, r, \Cchi = k)$ be the smallest defining number of all $r$-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 $r \geq 2(k-1)$ then $d(n, r, \Cchi = k)=k-1$. In this paper we show that for a given $k$ and for all $n < 3k$ and $r\geq 2(k-1)$, $d(n, r, \Cchi=k)=k-1$.