On the Minimum Number of Hamiltonian Cycles in Regular Graphs

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    A graph construction that produces a k-regular graph on n vertices for any choice of k ⩾ 3 and n = m(k + 1) for integer m ⩾ 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 ⩾ 5 and n ⩾ 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 ⩾ 11.

    Original languageEnglish
    Pages (from-to)426-430
    Number of pages5
    Issue number4
    Publication statusPublished - 2 Oct 2018


    • 05C30
    • 05C45
    • construction
    • Hamiltonian cycles
    • minimal
    • regular graphs


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