Distinguishing graphs by edge-colourings

We investigate the distinguishing index D′(G) of a graph G as the least number d such that G has an edge-colouring with d colours that is only preserved by the trivial automorphism. This is an analog to the notion of the distinguishing number D(G) of a graph G, which is defined for colourings of vertices. We obtain a general upper bound D′(G) ≤ ∆(G) unless G is a small cycle C3, C4 or C5. We also investigate the distinguishing chromatic index χD(G) defined for proper edge-colourings of a graph G. We prove that χD(G) ≤ ∆(G) + 1 except for four graphs C4, K4, C6 and K3,3. It follows that, surprisingly, each connected Class 2 graph G admits a minimal proper edge-colouring, i.e., with ∆(G) + 1 colours, preserved only by the trivial automorphism.