The distinguishing number and the distinguishing index of Cayley graphs

TL;DR

This paper investigates the minimum number of labels needed to uniquely identify Cayley graphs through vertex and edge labelings, focusing on their automorphism properties.

Contribution

It provides new insights into the distinguishing number and index specifically for Cayley graphs, expanding understanding of their symmetry-breaking labelings.

Findings

Determined bounds for the distinguishing number of certain Cayley graphs.

Established relationships between group properties and the distinguishing index.

Identified classes of Cayley graphs with minimal labelings for automorphism breaking.

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. In this paper, we investigate the distinguishing number and the distinguishing index of Cayley graphs.