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Transversal Hamilton cycles in digraph collections

Part of: Graph theory

Published online by Cambridge University Press:  27 March 2026

Yangyang Cheng ,
Heng Li ,
Wanting Sun  and
Guanghui Wang
Yangyang Cheng
Affiliation:
University of Oxford, UK
Heng Li
Affiliation:
Shandong University, China
Wanting Sun*
Affiliation:
Shandong University, China
Guanghui Wang
Affiliation:
Shandong University, China
*
Corresponding author: Wanting Sun; Email: wtsun@sdu.edu.cn
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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 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 generalisation of Ghouila-Houri’s theorem, thereby solving a problem proposed by Chakraborti, Kim, Lee, and Seo. Our proof utilises 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.

Keywords

TransversalHamilton cycleDigraph collectionSemi-degree

MSC classification

Primary: 05C35: Extremal problems

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Type
Paper
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Combinatorics, Probability and Computing , First View , pp. 1 - 57
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Copyright
© The Author(s), 2026. Published by Cambridge University Press

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