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On the minimum number of Hamiltonian cycles in regular graphs. (English) Zbl 1403.05082
Summary: A graph construction that produces a \(k\)-regular graph on \(n\) vertices for any choice of \(k\geq 3\) and \(n=m(k+1)\) for integer \(m\geq 2\) is described. The number of Hamiltonians cycles in such graphs can be explicitly determined as a function of \(n\) and \(k\), and empirical evidence is provided that suggests that this function gives a tight upper bound on the minimum number of Hamiltonian cycles in \(k\)-regular graphs on \(n\) vertices for \(k\) \(\geq 5\) and \(n\geq k+3\). An additional graph construction for 4-regular graphs is described for which the number of Hamiltonian cycles is superior to the above function in the case when \(k=4\) and \(n\geq 11\).

MSC:
05C45 Eulerian and Hamiltonian graphs
05C30 Enumeration in graph theory
Software:
GENREG
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References:
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