Transversal cycles and paths in tournaments

TL;DR

This paper extends classical results on Hamilton paths and cycles in large tournaments to transversal versions involving collections of tournaments, showing the existence of such structures with all orientations except possibly the consistent one.

Contribution

It introduces transversal Hamilton paths and cycles in collections of tournaments, generalizing Thomason's classical results and providing new proofs.

Findings

Existence of transversal Hamilton cycles with all orientations except possibly the consistent one.

Existence of transversal Hamilton paths with all orientations.

Generalization of Thomason's theorem to collections of tournaments.

Abstract

Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This paper establishes $\textit{transversal}$ generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on the common vertex set $V$, an $m$-edge directed subgraph $\mathcal{D}$ with the vertices in $V$ is called a transversal if there exists an bijection $\varphi\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{\varphi(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $n$, there exist transversal Hamilton cycles of all possible orientations possibly except the consistently oriented one. We also obtain a similar result for the transversal Hamilton paths of all possible orientations. These results generalize the classical theorem of Thomason, and our approach provides another proof of this theorem.