A bound for the distinguishing index of regular graphs

TL;DR

This paper proves that all connected finite regular graphs except K2 can be distinguished with three colours, and extends the result to certain infinite graphs, providing bounds on the distinguishing index.

Contribution

It establishes a bound of three colours for distinguishing edge-colourings of connected regular graphs, advancing the conjecture for finite and infinite cases.

Findings

All connected finite regular graphs except K2 admit a 3-colour distinguishing edge-colouring.

Infinite, locally finite graphs also admit such colourings.

Certain infinite regular graphs can be distinguished with only two colours.

Abstract

An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing edge-colouring with two colours. We show that all such graphs except $K_2$ admit a distinguishing edge-colouring with three colours. This result also extends to infinite, locally finite graphs. Furthermore, we are able to show that there are arbitrary large infinite cardinals $\kappa$ such that every connected $\kappa$-regular graph has distinguishing edge-colouring with two colours.