A spanning union of cycles in thin cylinder, torus and Klein bottle grid graphs

We propose an algorithm for obtaining the common transfer digraph $D^*_m$ for enumeration of 2-factors in graphs from the title all of which with $m n$ vertices ($m, n \in N, m >1$). The numerical data gathered for $m <19$ reveal some matchings of the numbers of 2-factors for different types of torus or Klein bottle. In latter case we conjecture that these numbers are invariant under twisting.