Common circulant homogeneous factorisations of the complete digraph

In this paper we determine the positive integers n and k for which there exists a homogeneous factorisation of a complete digraph on n vertices with k ‘common circulant’ factors. This means a partition of the arc set of the complete digraph K n into k circulant factor digraphs, such that a cyclic group of order n acts regularly on the vertices of each factor digraph whilst preserving the edges, and in addition, an overgroup of this permutes the factor digraphs transitively amongst themselves. This determination generalises a previous result for self-complementary circulants.