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A conjecture on the number of Hamiltonian cycles on thin grid cylinder graphs. (English) Zbl 1311.05102
Summary: We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph $$C_m \times P_{n+1}$$. We distinguish two types of Hamiltonian cycles, and denote their numbers $$h_m^A(n)$$ and $$h_m^B(n)$$. For fixed $$m$$, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we derive their generating functions and other related results for $$m\leq10$$. The computational data we gathered suggests that $$h^A_m(n)\sim h^B_m(n)$$ when $$m$$ is even.

##### MSC:
 05C45 Eulerian and Hamiltonian graphs 05C30 Enumeration in graph theory 05A15 Exact enumeration problems, generating functions
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