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

We propose an algorithm for obtaining the common transfer digraph $D_m^{\ast }$ for enumeration of 2-factors in graphs from the title all of which with $mn$ 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.