When the cartesian product of directed cycles is Hamiltonian

The cartesian product of two hamiltonian graphs is always hamiltonian. For directed graphs, the analogous statement is false. We show that the cartesian product C n 1 × C n 2 of directed cycles is hamiltonian if and only if the greatest common divisor (g.c.d.) d of n 1 and n 2 is at least two and there exist positive integers d 1 , d 2 so that d 1 + d 2 = d and g.c.d. ( n 1 , d 1 ) = g.c.d. ( n 2 , d 2 ) = 1. We also discuss some number‐theoretic problems motivated by this result.