The distinguishing index D′(G) of a graph G is the least cardinal d such that G has an edge colouring with d colours that is only preserved by the trivial automorphism. This is similar to the notion of the distinguishing number D(G) of a graph G, which is defined with respect to vertex colourings. We derive several bounds for infinite graphs, in particular, we prove the general bound D′(G) 6 ∆(...

A proper edge colouring of a graph G without isolated edges is neighbour-distinguishing if any two adjacent vertices have distinct sets consisting of colours of their incident edges. The neighbour-distinguishing index of G is the minimum number ndi(G) of colours in a neighbour-distinguishing edge colouring of G. According to a conjecture by Zhang, Liu and Wang (2002), ndi(G) ≤ ∆(G) + 2 provided...

It is proved that edges of a graph G can be coloured using χ(G) + 2 colours so that any two adjacent vertices have distinct sets of colours of their incident edges. In the case of a bipartite graph three colours are sufficient.