  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 $n \geq k \geq 4$. 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 $n$.
