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

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 $m$.}