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A limit conjecture on the number of Hamiltonian cycles on thin triangular grid cylinder graphs. (English) Zbl 1390.05099
Summary: We continue our research in the enumeration of Hamiltonian cycles (HCs) on thin cylinder grid graphs $$C_m\times P_{n_+1}$$ by studying a triangular variant of the problem. There are two types of HCs, distinguished by whether they wrap around the cylinder. Using two characterizations of these HCs, we prove that, for fixed $$m$$, the number of HCs of both types satisfy some linear recurrence relations. For small $$m$$, computational results reveal that the two numbers are asymptotically the same. We conjecture that this is true for all $$m\geq2$$.
##### MSC:
 05C30 Enumeration in graph theory 05C38 Paths and cycles 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05A15 Exact enumeration problems, generating functions
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