Hamilton Transversals in Tournaments

It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection \(\textbf{T}=(T_1,\dots ,T_m)\) of not-necessarily distinct tournaments on a common vertex set V, an m-edge directed graph \(\mathcal {D}\) with vertices in V is called a \(\textbf{T}\)-transversal if there exists a bijection \(\phi :E(\mathcal {D})\rightarrow [m]\) such that \(e\in E(T_{\phi (e)})\) for all \(e\in E(\mathcal {D})\). We prove that for sufficiently large m with \(m=|V|-1\), there exists a \(\textbf{T}\)-transversal Hamilton path. Moreover, if \(m=|V|\) and at least \(m-1\) of the tournaments \(T_1,\ldots ,T_m\) are assumed to be strongly connected, then there is a \(\textbf{T}\)-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub \(\textbf{H}\)-partition.