Homogeneous factorisations of graph products

A homogeneous factorisation of a digraph ΓΓ consists of a partition P={P1,…,Pk}P={P1,…,Pk} of the arc set AΓAΓ and two vertex-transitive subgroups M⩽G⩽Aut(Γ)M⩽G⩽Aut(Γ) such that M   fixes each PiPi setwise while G   leaves PP invariant and permutes its parts transitively. Given two graphs Γ1Γ1 and Γ2Γ2 we consider several ways of taking a product of Γ1Γ1 and Γ2Γ2 to form a larger graph, namely the direct product, cartesian product and lexicographic product. We provide many constructions which enable us to lift homogeneous factorisations or certain arc partitions of Γ1Γ1 and Γ2Γ2, to homogeneous factorisations of the various products.