 
Summary: On the Maximum Number of Hamiltonian Paths in
Tournaments
Ilan Adler
Noga Alon
Sheldon M. Ross
August 2000
Abstract
By using the probabilistic method, we show that the maximum number of di
rected Hamiltonian paths in a complete directed graph with n vertices is at least
(e  o(1)) n!
2n1 .
1 Introduction
A tournament T is an oriented complete graph. A Hamiltonian path in T is a spanning
directed path in it. Let P(T) denote the number of Hamiltonian paths in T. For n 2,
define P(n) = max P(T), where T ranges over all tournaments T on n vertices. More
than 50 years ago, Szele [5] showed that
P(n)
n!
2n1
(1)
