Distinguishing Number for some Circulant Graphs

TL;DR

This paper investigates the distinguishing number of circulant graphs, providing a construction for a family with unique distinguishing numbers that are independent of the order n, expanding understanding of symmetry-breaking in these graphs.

Contribution

The paper introduces a new construction of circulant graphs with distinct, order-independent distinguishing numbers, advancing the study of graph automorphisms and symmetry.

Findings

Constructed a family of circulant graphs with unique distinguishing numbers

Showed that these distinguishing numbers do not depend on the graph order n

Extended understanding of symmetry-breaking in circulant graphs

Abstract

Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs depending on $n$. In this paper, we study circulant graphs of order $n$ where the adjacency is defined using a symmetric subset $A$ of $\mathbb{Z}_n$, called generator. We give a construction of a family of circulant graphs of order $n$ and we show that this class has distinct distinguishing numbers and these lasters are not depending on $n$.