The Structure of the 2-factor Transfer Digraph common for Thin Cylinder, Torus and Klein Bottle Grid Graphs

TL;DR

This paper analyzes the structure of the transfer digraph used for counting 2-factors in certain grid graphs, revealing its component structure varies with the parity of the width and connecting it to recent results on related graphs.

Contribution

It characterizes the component structure of the transfer digraph for thin cylinder, torus, and Klein bottle grid graphs, extending previous results to new graph classes.

Findings

For odd m, the transfer digraph has two isomorphic components of size 2^{m-1}.

For even m, it has floor(m/2)+1 components with sizes expressed via binomial coefficients.

Most components are bipartite digraphs, except one.

Abstract

We prove that the transfer digraph ${\cal D}^*_{C,m}$ needed for the enumeration of 2-factors in the thin cylinder $TnC_{m}(n)$, torus $TG_{m}(n)$ and Klein bottle $KB_m(n)$ (all grid graphs of the fixed width $m$ and with $m \cdot n$ vertices), when $m$ is odd, has only two components of order $2^{m-1}$ which are isomorphic. When $m$ is even, ${\cal D}^*_{C,m}$ has $ \left\lfloor \frac{m}{2} \right\rfloor + 1$ components which orders can be expressed via binomial coefficients and all but one of the components are bipartite digraphs. The proof is based on the application of recently obtained results concerning the related transfer digraph for linear grid graphs (rectangular, thick cylinder and Moebius strip).