A spanning union of cycles in rectangular grid graphs, thick grid cylinders and Moebius strips

TL;DR

This paper introduces an algorithm to analyze spanning unions of cycles in various grid graphs, providing enumeration methods and revealing properties of 2-factors in these structures.

Contribution

It develops a transfer digraph approach for enumerating 2-factors in grid graphs, extending understanding of Hamiltonian cycle generalizations in these graphs.

Findings

Algorithm for constructing transfer digraphs ${\

Properties of digraphs ${\

Enumeration sequences for 2-factors in grid graphs

Abstract

Motivated to find the answers to some of the questions that have occurred in recent papers dealing with Hamiltonian cycles (abbreviated HCs) in some special classes of grid graphs we started the investigation of spanning unions of cycles, the so-called 2-factors, in these graphs (as a generalizations of HCs). For all the three types of graphs from the title and for any integer $m \geq 2$ we propose an algorithm for obtaining a specially designed (transfer) digraph ${\cal D}^*_m$. The problem of enumeration of 2-factors is reduced to the problem of enumerating oriented walks in this digraph. Computational results we gathered for $m \leq 17$ reveal some interesting properties both for the digraphs ${\cal D}^*_m$ and for the sequences of numbers of 2-factors. We prove some of them for arbitrary $m \geq 2$.