Hamilton transversals in tournaments

TL;DR

This paper extends classical Hamilton path and cycle results in tournaments to transversal versions involving collections of tournaments, proving the existence of such paths and cycles under certain conditions.

Contribution

It introduces the concept of $ extbf{T}$-transversals and proves their existence for Hamilton paths and cycles in large collections of tournaments.

Findings

Existence of $ extbf{T}$-transversal Hamilton paths for large collections.

Existence of $ extbf{T}$-transversal Hamilton cycles when most tournaments are strongly connected.

Introduction of $ extbf{H}$-partition technique for tournament analysis.

Abstract

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 $\mathbf{T}=\{T_1,\dots,T_m\}$ 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 $\mathbf{T}$-transversal if there exists a bijection $\phi\colon E(\mathcal{D})\to [m]$ 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 $\mathbf{T}$-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 $\mathbf{T}$-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub $\mathbf{H}$-partition.