Distinguishing regular graphs from lists

TL;DR

This paper proves that all sufficiently large connected regular graphs, including infinite ones, can be edge-colour distinguished from lists of size two, extending to certain infinite cardinalities.

Contribution

It establishes that every large connected regular graph admits a list edge colouring that is distinguishing, generalizing previous results to infinite and larger graphs.

Findings

All connected regular graphs of order at least 7 have a 2-list distinguishing edge colouring.

The result extends to regular graphs of fixed points in the aleph hierarchy.

The theorem applies to both finite and certain infinite regular graphs.

Abstract

An edge colouring of a graph is called distinguishing if there is no non-trivial automorphism which preserves it. We prove that every at most countable, finite or infinite, connected regular graph of order at least $7$ admits a distinguishing edge colouring from any set of lists of length $2$. Furthermore, we show that the same holds for connected regular graphs of order $\kappa$ where $\kappa$ is a fixed point of the aleph hierarchy.