Some equitably 2-colourable cycle decompositions of multipartite graphs - Waterhouse, Mary (Utilitas Mathematica, 2006)
Let G be a graph in which each vertex has been coloured using one of k colours, say c1, c2, . . . , ck. If an m-cycle C in G has xi vertices coloured ci, i = 1, 2, . . . , k, and |xi - xj| ≤ 1 for any ci, cj ∈ c1, c2, . . . , ck, then C is said to be equitably k-coloured. An m-cycle decomposition C of a graph G is equitably k-colourable if the vertices of G can be coloured so that every m-cycle in C is equitably k-coloured. For m = 3 and 5 we completely settle the existence problem for equitably 2-colourable m-cycle decompositions of complete multipartite graphs with all parts the same size, while for m = 4 and 6 we settle the same problem for all complete multipartite graphs.