Constructing arbitrarily large graphs with a specified number of Hamiltonian cycles

Michael Haythorpe (School of Computer Science, Engineering and Mathematics, Flinders University, 1284 South Road, Clovelly Park, SA 5042, Australia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 4, No 1 (2016): Electronic Journal of Graph Theory and Applications

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 $n$.