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Transversal Hamilton cycles in digraph collections

Yangyang Cheng, Heng Li, Wanting Sun, Guanghui Wang

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 generalization of Ghouila-Houri's theorem, thereby solving a problem proposed by Chakraborti, Kim, Lee and Seo \cite{2023Tournament}. Our proof utilizes the absorption method for transversals, the regularity method for digraph collections, as well as the transversal blow-up lemma \cite{cheng2023transversals} 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 \cite{2021jooskim}.

Transversal Hamilton cycles in digraph collections

Abstract

Given a collection of digraphs on the common vertex set , an -edge digraph with vertices in is transversal in if there exists a bijection such that for all . Ghouila-Houri proved that any -vertex digraph with minimum semi-degree at least 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 \cite{2023Tournament}. Our proof utilizes the absorption method for transversals, the regularity method for digraph collections, as well as the transversal blow-up lemma \cite{cheng2023transversals} and the related machinery. As an application, when is sufficiently large, our result implies the transversal version of Dirac's theorem, which was proved by Joos and Kim \cite{2021jooskim}.
Paper Structure (19 sections, 28 theorems, 114 equations, 5 figures)

This paper contains 19 sections, 28 theorems, 114 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Notation and Organization
  4. Discussion of the strategy
  5. Sketch of the proof of Theorem \ref{['mainthm']}
  6. Preliminaries
  7. Regularity for digraph collections
  8. Transversal perfect matching in bipartite graph collections
  9. Probabilistic tool
  10. Absorbing
  11. Stable case
  12. Extremal case
  13. Proof of Theorem \ref{['theorem-EC12']}
  14. Proof of Theorem \ref{['theorem-EC3']}
  15. Concluding remarks
  16. ...and 4 more sections

Key Result

Theorem 1.2

Suppose $\mathcal{G} = \{G_1, \ldots, G_n\}$ is a collection of graphs with the same vertex set of size $n$. If $\delta(G_i)\geq \frac{n}{2}$ for all $i\in [n]$, then $\mathcal{G}$ contains a transversal Hamilton cycle.

Figures (5)

  • Figure 1: Extremal digraphs EC1, EC2, and EC3. The gray shaded elliptical indicates that the digraph induced by this vertex set is complete, the gray shaded arrow between two vertex sets indicates that the induced digraph by them is complete in this direction.
  • Figure 2: A Type-I directed $c$-absorbing path of $(v,v)$ and a Type-I directed $c$-absorbing path of $(v,u)$ with $v\neq u$.
  • Figure 3: A Type-II directed $c$-absorbing path of $(v,v)$ and a Type-II directed $c$-absorbing path of $(v,u)$ with $v\neq u$.
  • Figure 4: Absorbing.
  • Figure 5: $w_1w_2$ is a $c$-absorbing edge of $(u, v)$.

Theorems & Definitions (87)

  • Definition 1.1
  • Theorem 1.2: 2021jooskim
  • Theorem 1.3: Houri
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2: Regularity lemma for digraph collections
  • Definition 2.3: Reduced digraph collection
  • Lemma 2.4: Degree inheritance
  • proof
  • Theorem 2.5: bradshaw2021transversals
  • ...and 77 more