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

In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HCs) 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 HCs. Here, in Part~II, we proceed further on to the contractible case. We propose two different novel characterizations of contractible HCs, from which we construct digraphs for enumerating the contractible HCs. Given the impression which the computational data for $m leq 9$ convey, we conjecture that the asymptotic domination of the contractible HCs versus the non-contractible HCs, among the total number of HCs, depends on the parity of $m$.}