Edge motion and the distinguishing index

The distinguishing index D′(G) of a graph G is the least number d such that G has an edge colouring with d colours that is preserved only by the trivial automorphism. We investigate the edge motion of a graph with respect to its automorphisms and compare it with the vertex motion. We prove an analog of the Motion Lemma of Russell and Sundaram, and we use it to determine the distinguishing index of powers of complete graphs and of cycles with respect to the Cartesian, direct and strong product.