A set of tournaments with many Hamiltonian cycles
Hayato Ushijima‐Mwesigwa · TigerPrints (Clemson University) · 2012
For a random tournament on $3^n$ vertices, the expected number of Hamiltonian cycles is known to be $(3^n -1)!/2^{3^n}$. Let $T_1$ denote a tournament of three vertices $ {v_1, v_2, v_3}$. Let the orientation be such that there are directed edges from $v_1 $to $v_2$ , from $v_2$ to $v_3$ and from $v_3$ to $ v_1$. Construct a tournament $T_i$ by making three copies of $T_{i-1}$, $T_{i-1}'$, $T_{i-1}''$ 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.