On the Minimum Number of Hamiltonian Cycles in Regular Graphs

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.