Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles - Michael Haythorpe (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 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.