Vertex colouring edge partitions

A partition of the edges of a graph G into sets { S 1 , … , S k } defines a multiset X v for each vertex v where the multiplicity of i in X v is the number of edges incident to v in S i . 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 , X u ≠ X v . 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.