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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 togrow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractibleHC's. Here, in Part~II, we proceed further on to the contractible case. Wepropose two different novel characterizations of contractible HC's, from whichwe construct digraphs for enumerating the contractible HC's. Given theimpression which the computational data for $m \leq 9$ convey, we conjecturethat the asymptotic domination of the contractible HC's versus thenon-contractible HC's, among the total number of HC's, depends on the parity of$m$.}

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