Transversal cycles and paths in 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.