A bound for the distinguishing index of regular graphs

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 edgecolouring 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 κ such that every connected κ-regular graph has distinguishing edge-colouring with two colours.