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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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