A spanning union of cycles in thin cylinder, torus and Klein bottle grid graphs

TL;DR

This paper introduces an algorithm to enumerate 2-factors in grid graphs on cylinders, tori, and Klein bottles, revealing invariance properties and matching patterns across different topologies.

Contribution

It presents a novel algorithm for constructing the transfer digraph for 2-factor enumeration in complex grid graphs with various topologies.

Findings

Numerical data for m<19 show matching 2-factor counts across different topologies.

Conjecture that 2-factor counts are invariant under twisting in Klein bottle graphs.

Algorithm enables systematic enumeration of 2-factors in these grid graphs.

Abstract

We propose an algorithm for obtaining the common transfer digraph $ D^*_m$ for enumeration of 2-factors in graphs from the title all of which with $m n$ vertices ($m, n \in N, m >1 $). The numerical data gathered for $m <19$ reveal some matchings of the numbers of 2-factors for different types of torus or Klein bottle. In latter case we conjecture that these numbers are invariant under twisting.