The Structure of the 2-factor Transfer Digraph Common for Rectangular, Thick Cylinder and Moebius Strip Grid Graphs

In this paper, we prove that all but one of the components of the transfer digraph ${\cal D}^*_m$ needed for the enumeration of 2-factors in the rectangular, thick cylinder and Moebius strip grid graphs of the fixed width $m$ $(m \in N)$ are bipartite digraphs and that their orders could be expressed in term of binomial coefficients. In addition, we prove that the set of vertices of each component consists of all the binary $m$-words for which the difference of numbers of zeros in odd and even positions is constant.