The object of interest is the maximum number, h(n), of Hamilton cycles in an n-tournament. By considering the expected number of Hamilton cycles in various classes of random tournaments, we obtain new asymptotic lower bounds on h(n). The best result so far is approximately 2.85584... times the expected number g(n) of Hamilton cycles in a random n-tournament, and it is conjectured that h(n) ∼ cg(n) where c ≈ 2.855958. The same statements hold for Hamilton paths. AMS Classifications: 05A16, 05C30… CONTINUE READING