A Conjecture on the Number of Hamiltonian Cycles on Thin Grid Cylinder Graphs

We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph Cm × Pn+1. We distinguish two types of Hamiltonian cycles depending on their contractibility (as Jordan curves) and denote their numbers h m (n) and hm(n). For fixedm, both of them satisfy linear homogeneous recurrence relations with constant coefficients. We derive their generating functions and other related results for m ≤ 10. The computational data we gathered suggests that h m (n) ∼ hm(n) when m is even.