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Hamiltonicity after reversing the directed edges at a vertex of a Cartesian product

Let $\vec C_m$ and $\vec C_n$ be directed cycles of length $m$ and $n$, with $m,n \ge 3$, and let $P(\vec C_m \mathbin{\Box} \vec C_n)$ be the digraph that is obtained from the Cartesian product $\vec C_m \mathbin{\Box} \vec C_n$ by choosing a vertex $v$, and reversing the orientation of all four directed edges that are incident with $v$. (This operation is called "pushing" at the vertex $v$.) By applying a special case of unpublished work of S.X.Wu, we find elementary number-theoretic necessary and sufficient conditions for the existence of a hamiltonian cycle in $P(\vec C_m \mathbin{\Box} \vec C_n)$. A consequence is that if $P(\vec C_m \mathbin{\Box} \vec C_n)$ is hamiltonian, then $\gcd(m,n) = 1$, which implies that $\vec C_m \mathbin{\Box} \vec C_n$ is not hamiltonian. This final conclusion verifies a conjecture of J.B.Klerlein and E.C.Carr.