Distinguishing regular graphs from lists

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 $\hat{I}º$ where $\hat{I}º$ is a fixed point of the aleph hierarchy.