When the cartesian product of two directed cycles is hyperhamiltonian

We say a digraph G is hyperhamiltonian if there is a spanning closed walk in G which passes through one vertex exactly twice and all others exactly once. We show the cartesian product Z_a × Z_b of two directed cycles is hyperhamiltonian if and only if there are positive integers m and n with ma + nb = ab + 1 and gcd(m, n) = 1 or 2. We obtain a similar result for the vertex-deleted subdigraphs of Z_a × Z_b.