Vertex colouring edge partitions

A partition of the edges of a graph G into sets {S1,…,Sk} defines a multiset Xv for each vertex v where the multiplicity of i in Xv is the number of edges incident to v in Si. We show that the edges of every graph can be partitioned into 4 sets such that the resultant multisets give a vertex colouring of G. In other words, for every edge (u,v) of G, Xu≠Xv. Furthermore, if G has minimum degree at least 1000, then there is a partition of E(G) into 3 sets such that the corresponding multisets yield a vertex colouring.