The Tournament Theorem of R\'edei revisited

Abstract

In 1934 L. R\'edei published his famous theorem that the number of Hamiltonian paths in a tournament is odd. In fact it is a corollary of a stronger theorem in his paper. Stronger theorems were also obtained in the early 1970s by G.A. Dirac in his lectures at Aarhus University and by C. Berge in his monographs on graphs and hypergraphs. We exhibit the stronger theorems of R\'edei, Dirac and Berge and explain connections between them. The stronger theorem of Dirac has two corollaries, one equivalent to R\'edei's stronger theorem and the other related to Berge's stronger theorem.