Slater orders and Hamiltonian paths of tournaments

A Slater order of a tournament T is a linear order at minimum distance with respect to the number of arcs that must be reversed in T to make T transitive. We compare here the number of Slater orders that a tournament can have with its number of Hamiltonian paths. More precisely, we specify a lower bound (got from some special tournaments) and an upper bound (given by the maximum number of Hammiltonian paths that a tournament can have) of the maximum number of Slater orders that a tournament can have. Then, we study two special classes of tournaments for which we compute or estimate the number of Slater orders and Hamiltonian paths.