  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
Hamiltonian 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.
