Enumeration of Hamiltonian Cycles on a Thick Grid Cylinder -- Part II: Contractible Hamiltonian Cycles

TL;DR

This paper advances the enumeration of Hamiltonian cycles on grid cylinder graphs by focusing on contractible cycles, introducing new characterizations and digraph constructions, and analyzing their asymptotic behavior relative to non-contractible cycles.

Contribution

It introduces two novel characterizations of contractible Hamiltonian cycles and constructs digraphs for their enumeration, expanding understanding from previous work on non-contractible cycles.

Findings

Contractible HC's dominate non-contractible HC's asymptotically for certain m.

New characterizations enable systematic enumeration of contractible HC's.

Computational data up to m=9 supports conjectures on cycle dominance based on parity.

Abstract

In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HC's) on the grid cylinder graphs $P_{m+1}\times C_n$, where $n$ is allowed to grow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractible HC's. Here, in Part~II, we proceed further on to the contractible case. We propose two different novel characterizations of contractible HC's, from which we construct digraphs for enumerating the contractible HC's. Given the impression which the computational data for $m \leq 9$ convey, we conjecture that the asymptotic domination of the contractible HC's versus the non-contractible HC's, among the total number of HC's, depends on the parity of $m$.}