An edge colouring of multigraphs

We consider a strict k-colouring of a multigraph G as a surjection f from the vertex set of G into a set of colours {1,2,. . . ,k} such that, for every non-pendant vertex x of G, there exist at least two edges incident to x and coloured by the same colour. The maximum number of colours in a strict edge colouring of G is called the upper chromatic index of G and is denoted by χ(G). In this paper we prove some results about it.