Homogeneous factorisations of graphs and digraphs

A homogeneous factorisation (M,G,Γ,P) is a partition P of the arc set of a digraph Γ such that there exist vertex transitive groups M < G 6 Aut(Γ) such that M fixes each part of P setwise while G acts transitively on P. Homogeneous factorisations of complete graphs have previously been studied by the second and fourth authors, and are a generalisation of vertex-transitive self-complementary digraphs. In this paper we initiate the study of homogeneous factorisations of arbitrary graphs and digraphs. We give a generic group theoretic construction and show that all homogeneous factorisations can be constructed in this way. We also show that the important homogeneous factorisations to study are those where G acts transitively on the set of arcs of Γ, M is a normal subgroup of G and G/M is a cyclic group of prime order.