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Summary: The Maximum Number of Hamiltonian Paths in Tournaments
Noga Alon*
Raymond and Beverly Sackler Faculty of Exact Sciences
School of Mathematical Sciences
Tel Aviv University
Ramat Aviv, Tel-Aviv, Israel
* Research supported in part by a U.S.A.-Israel BSF grant and by a Bergmann Memorial Grant.
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Abstract.
Solving an old conjecture of Szele we show that the maximum number of directed Hamiltonian
paths in a tournament on n vertices is at most c · n3/2
· n!
2n-1 , where c is a positive constant
independent of n.
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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 on n vertices. More than forty years ago,
Szele [Sz] showed that
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