Transversal Hamilton cycles in digraph collections

TL;DR

This paper generalizes Ghouila-Houri's theorem to transversal Hamilton cycles in collections of digraphs, using advanced combinatorial methods, and extends to a transversal version of Dirac's theorem for large graphs.

Contribution

It provides a transversal Hamilton cycle existence theorem in digraph collections, solving a problem by Chakraborti et al., and introduces new proof techniques.

Findings

Established a transversal Hamilton cycle theorem for large digraph collections.

Extended Dirac's theorem to a transversal setting for sufficiently large graphs.

Abstract

Given a collection $\mathcal{D} =\{D_1,D_2,\ldots,D_m\}$ of digraphs on the common vertex set $V$, an $m$-edge digraph $H$ with vertices in $V$ is \textit{transversal} in $\mathcal{D}$ if there exists a bijection $\varphi :E(H)\rightarrow [m]$ such that $e \in E(D_{\varphi(e)})$ for all $e\in E(H)$. Ghouila-Houri proved that any $n$-vertex digraph with minimum semi-degree at least $\frac{n}{2}$ contains a directed Hamilton cycle. In this paper, we provide a transversal generalization of Ghouila-Houri's theorem, thereby solving a problem proposed by Chakraborti, Kim, Lee and Seo. Our proof utilizes the absorption method for transversals, the regularity method for digraph collections, as well as the transversal blow-up lemma and the related machinery. As an application, when $n$ is sufficiently large, our result implies the transversal version of Dirac's theorem, which was proved by Joos and Kim.