A bound for the distinguishing index of regular graphs. (October 2020) - Lehner, Florian; Pilśniak, Monika; Stawiski, Marcin
Abstract: An edge-colouring of a graph is distinguishing if the only automorphism that 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 κ such that every connected κ -regular graph has a distinguishing edge-colouring with two colours.