Homogeneous factorisations of complete multipartite graphs

A homogeneous factorisation of a graph is a partition of its arc set such that there exist vertex transitive subgroups M < G 6 Aut(Γ) with M fixing each part of the partition setwise and G preserving the partition and transitively permuting the parts. In this paper we study homogeneous factorisations of complete multipartite graphs such that M acts regularly on vertices. We provide a necessary and sufficient condition for the existence of such factorisations and produce many interesting examples. In particular we give a complete determination of the parameters for which homogeneous factorisations exist with M cyclic.