Distinguishing actions of symmetric groups and related graphs

TL;DR

This paper determines the distinguishing numbers for all actions of symmetric groups and related graphs, solving several open problems and advancing understanding of symmetry-breaking in combinatorial structures.

Contribution

It provides a complete characterization of the distinguishing numbers for all symmetric group actions and associated graphs, addressing previously unresolved questions.

Findings

Explicit formulas for distinguishing numbers of symmetric group actions

Complete classification of graphs with automorphism groups isomorphic to symmetric groups

Resolution of multiple open problems in symmetry-breaking literature

Abstract

The distinguishing number $D(G,X)$ of an action of a group $G$ on a set $X$ is the least size of a partition of $X$ such that no element of $G$ acting nontrivially on $X$ preserves this partition. In this paper we describe the distinguishing numbers for all actions of the symmetric group $S_n$, for any $n\geq 3$. This allows us to describe the distinguishing numbers for all graphs whose automorphism group is isomorphic with a symmetric group. Our description solves a few open problems posed by various authors in earlier papers on this topic.