I have two circulant Cayley digraphs: that is, Cayley digraphs X = Cay(ℤ/m, S) and Y = Cay(ℤ/n, T), for odd integers m < n, and sets with sizes |S| = (m − 1)/2, and |T| = (n − 1)/2.
These digraphs are antisymmetric, in that S is disjoint from −S, and T is disjoint from −T. (It follows that for each distinct pair of vertices a,b in either graph, there is either an arc from a to b, or vice versa.)
Question. What conditions on m, n, S, and T must hold for X to be an induced directed subgraph of Y?