  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.

Combinatorics