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Cycle tilings and $H$-factors in directed graphs

Theodore Molla, Andrew Treglown

Abstract

We prove several results concerning cycle tilings and $H$-factors in digraphs. We provide a minimum semi-degree condition for forcing a digraph to contain a given spanning collection of vertex-disjoint orientations of cycles. Our result is asymptotically best possible for odd cycles and can be viewed as a digraph analogue of the El-Zahar conjecture. In addition, we asymptotically determine the minimum degree threshold for forcing an $H$-factor in a digraph for a range of digraphs $H$, including the cases when $H$ is a tree or anti-directed cycle. Furthermore, an asymptotically exact Ore-type result for forcing a transitive tournament factor in a digraph is proven. Several related open problems are also highlighted.

Cycle tilings and $H$-factors in directed graphs

Abstract

We prove several results concerning cycle tilings and -factors in digraphs. We provide a minimum semi-degree condition for forcing a digraph to contain a given spanning collection of vertex-disjoint orientations of cycles. Our result is asymptotically best possible for odd cycles and can be viewed as a digraph analogue of the El-Zahar conjecture. In addition, we asymptotically determine the minimum degree threshold for forcing an -factor in a digraph for a range of digraphs , including the cases when is a tree or anti-directed cycle. Furthermore, an asymptotically exact Ore-type result for forcing a transitive tournament factor in a digraph is proven. Several related open problems are also highlighted.
Paper Structure (18 sections, 25 theorems, 72 equations)

This paper contains 18 sections, 25 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Minimum degree conditions forcing $H$-factors
  3. Cycle tilings and an analogue of the El-Zahar conjecture
  4. Ore-type results for transitive tournament factors
  5. Useful results
  6. The diregularity lemma
  7. The absorbing lemma
  8. Probabilistic estimates
  9. Proof of Theorem \ref{['mainthm']}
  10. Proof of the EL-Zahar-type result
  11. Auxiliary results for the proof of Theorem \ref{['thm:elzthm']}
  12. An absorbing lemma for Theorem \ref{['thm:elzthm']}
  13. Proof of Theorem \ref{['thm:elzthm']}
  14. Proof of Theorem \ref{['orethm']}
  15. Proof of Theorem \ref{['orethm2']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

hs Let $n \in \mathbb N$ be divisible by $r \in \mathbb N$. If $G$ is an $n$-vertex graph with $\delta (G) \geq (1-1/r)n$, then $G$ contains a $K_r$-factor. Moreover, the bound on $\delta (G)$ is tight.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 39 more