Non‐Hamiltonian Cartesian products of two even dicycles

In this note it is proven that there exist infinitely many positive integers and such that the Cartesian product of a directed cycle of length and a directed cycle of length is non‐Hamiltonian. In particular, the Cartesian product of an 880‐dicycle and a 4368‐dicycle is non‐Hamiltonian. We also prove that there is no such graph on fewer than vertices, which is rather astonishing.