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

TL;DR

This paper characterizes the structure of the transfer digraphs used for counting 2-factors in certain grid graphs, revealing bipartite components and their vertex sets based on binary word properties.

Contribution

It proves that most components of the transfer digraph are bipartite and their sizes relate to binomial coefficients, and describes their vertex sets in terms of binary words.

Findings

Most components are bipartite digraphs.

Component sizes are expressed via binomial coefficients.

Vertices correspond to binary words with fixed zero count differences.

Abstract

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.