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 ⩽ 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.