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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
##### Keywords:
construction; Hamiltonian cycles; minimal; regular graphs
GENREG
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