Hamilton Transversals in Tournaments

Abstract

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