A short note on the order of the double reduced 2-factor transfer digraph for rectangular grid graphs

TL;DR

This paper determines the exact order of a specific transfer digraph used for enumerating cycle unions in rectangular grid graphs when the grid width is odd, providing a precise combinatorial formula.

Contribution

It establishes a closed-form formula for the order of the double reduced 2-factor transfer digraph in the context of rectangular grid graphs with odd width.

Findings

Derived an explicit formula for the digraph's order.

Connected the digraph's order to binomial coefficients.

Enhanced understanding of cycle enumeration in grid graphs.

Abstract

We prove that the order of the double reduced 2-factor transfer digraph ${\cal R}^{**}_{m}$ which is needed for the enumeration of the spanning unions of cycles in the rectangular grid graph $P_m \times P_n$ ($m,n \in N$), when $m$ is odd, is equal to $\displaystyle \mid V({\cal R}^{**}_{m}) \mid = \frac{1}{2} \left[{m+1 \choose (m-1)/2 } + {(m+1)/2 \choose \lfloor (m+1)/4 \rfloor}\right].$