Distinguishing graphs by edge-colourings

We also introduce the distinguishing chromatic index χD′(G) defined for proper edge-colourings of a graph G. A correlation with distinguishing vertices by colour walks introduced in Kalinowski et al. (2004) is shown. We prove that χD′(G)≤Δ(G)+1 except for four small graphs C4, K4, C6 and K3,3. It follows that each connected Class 2 graph G admits a minimal proper edge-colouring, i.e., with χ′(G) colours, preserved only by the trivial automorphism.