The Tournament Theorem of Rédei revisited

TL;DR

Let $T$ be a tournament with at least $2$ vertices. The paper revisits Rédei's classical result on an odd number of Hamiltonian paths by presenting and connecting stronger parity theorems due to Rédei, Dirac, and Berge, and by showing how Dirac's Corollaries imply Rédei's theorem while Berge's results extend this parity framework. It formalizes Dirac's Stronger Theorem for mixed graphs, derives key corollaries, and demonstrates the equivalence of Rédei's Stronger Theorem with Dirac's Corollary 3. It then unifies Berge's Stronger Theorem and the Berge-Dirac Theorem, showing how parity of permutation-avoidance classes governs Hamiltonian-path counts under edge-type changes. The work therefore provides a cohesive view of parity-preserving transformations in tournaments and related mixed graphs and clarifies the exact implications among these classical strengthenings.

Abstract

In 1934 L. Rédei 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 Rédei, Dirac and Berge and explain connections between them. The stronger theorem of Dirac has two corollaries, one equivalent to Rédei's stronger theorem and the other related to Berge's stronger theorem.

Figures (2)

• Figure 1: A mixed graph $G$ with $n=3$ vertices.
• Figure 2: The sets $\mathbb{P}_i \subseteq \mathbb{P}(G)$ with $i \in \{0,1,2,3\}$.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['theorem:diracstrong']}
• Theorem 2.3
• Theorem 2.4
• Theorem 4.1