Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles - Michael Haythorpe (Electronic Journal of Graph Theory and Applications (EJGTA), 2016)
A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing $5n-9$ vertices and $k$ Hamiltonian cycles for any choice of integers $n ≥ k ≥ 4$. The construction is not designed to be minimal in any sense, but rather to ensure that the graphs produced remain non-trivial instances of the Hamiltonian cycle problem even when $k$ is chosen to be much smaller than $n$.