Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles

A constructive method is provided that outputs a directed graph which is named a broken crowngraph, 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 graphsproduced remain non-trivial instances of the Hamiltonian cycle problem even when k is chosen tobe much smaller than n.