On symmetries of edge and vertex colourings of graphs

Let c and c ^′ be edge or vertex colourings of a graph G. The stabiliser of c is the set of automorphisms of G that preserve the colouring. We say that c ^′ is less symmetric than c if the stabiliser of c ^′ is contained in the stabiliser of c. We show that if G is not a bicentred tree, then for every vertex colouring of G there is a less symmetric edge colouring with the same number of colours. On the other hand, if T is a tree, then for every edge colouring there is a less symmetric vertex colouring with the same number of colours. Our results can be used to characterise those graphs whose distinguishing index is larger than their distinguishing number.