Distinguishing numbers of finite $4$-valent vertex-transitive graphs

TL;DR

This paper determines the distinguishing numbers of finite 4-valent vertex-transitive graphs, showing that most have a distinguishing number of 2, with only a few exceptions including an infinite family.

Contribution

It provides a complete classification of the distinguishing numbers for finite 4-valent vertex-transitive graphs, identifying exceptions and establishing that most have distinguishing number 2.

Findings

Most finite 4-valent vertex-transitive graphs have distinguishing number 2.

There is one infinite family of such graphs with a different distinguishing number.

Finitely many examples also deviate from the typical distinguishing number of 2.

Abstract

The distinguishing number of a graph $G$ is the smallest $k$ such that $G$ admits a $k$-colouring for which the only colour-preserving automorphism of $G$ is the identity. We determine the distinguishing number of finite $4$-valent vertex-transitive graphs. We show that, apart from one infinite family and finitely many examples, they all have distinguishing number $2$.