A set of tournaments with many Hamiltonian cycles

For a random tournament on 3n vertices, the expected number of Hamiltonian cycles is known to be (3n− 1)!/23 . Let T1 denote a tournament of three vertices v1,v2,v3. Let the orientation be such that there are directed edges from v1to v2 , from v2 to v3 and from v3 to v1. Construct a tournament Ti by making three copies of Ti−1, T ′ i−1, T ′′ i−1 and T ′′′ i−1. Let each vertex in T ′ i−1 have directed edges to all vertices in T ′′ i−1, similarly place directed edges from each vertex in T ′′ i−1 to all vertices in T ′′′ i−1 and from T ′′′ i−1 to T ′ i−1. In this thesis, we shall study this family of highly symmetric tournaments. In particular we shall present two different algorithms to calculate the number of Hamiltonian cycles in these tournaments and compare them with the expected number and with known bounds for random tournaments. This thesis is motivated by the question of the maximum number of Hamiltonian cycles a tournament can have.