Hamiltonicity of the cartesian product of two directed cycles minus a subgroup

Let (Za ×Zb)�(Zc ×Zd) be the product of two directed cycles minus a subgroup. Also, let A = a and B = b . We show that this digraph is Hamiltonian if and only if there exist positive integers m and n such that Am + Bn= AB � 1 and gcd(dm,cn )=1 .