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On the Turánability and tileability of oriented graphs

Igor Araujo, Zimu Xiang

Abstract

An oriented graph $H$ is Turánable (resp. tileable) if there exist $n_0 \in \mathbb{N}$ such that every semi-regular near-tournament on $n \ge n_0$ vertices contains a copy of $H$ (resp. a perfect $H$-tiling). We disprove a conjectured characterization of Turánable oriented graphs by DeBiasio, Han, Lo, Molla, Piga, and Treglown, show that there are Turánable oriented graphs which are not tileable, and provide a new example of tileable oriented graph.

On the Turánability and tileability of oriented graphs

Abstract

An oriented graph is Turánable (resp. tileable) if there exist such that every semi-regular near-tournament on vertices contains a copy of (resp. a perfect -tiling). We disprove a conjectured characterization of Turánable oriented graphs by DeBiasio, Han, Lo, Molla, Piga, and Treglown, show that there are Turánable oriented graphs which are not tileable, and provide a new example of tileable oriented graph.

Paper Structure

This paper contains 15 sections, 30 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Turánable oriented graph not contained in $D_s$
  3. The (non)tileability of $D_s$
  4. Minimum semi-degree threshold for $C_6^2$
  5. The tileability of $D_{1,1,2}$
  6. Notation
  7. Almost perfect tiling via regularity
  8. Proof of Theorem \ref{['thm:d112']}
  9. Non-extremal case
  10. Extremal case
  11. Concluding remarks
  12. Turánability
  13. Tileability
  14. Semi-degree thresholds
  15. Proof of Lemma \ref{['lem:merge']}

Key Result

Theorem 1.6

There exists a Turánable oriented graph which is not a subgraph of $D_s$ for any $s \in \mathbb{N}$.

Figures (3)

  • Figure 1: Different drawings of the oriented graph $S$. Note that $S \subset C_5^2$, $S \subset F_2$, and $D_{1,1,2} \subset S$.
  • Figure 2: Copies of $D_2$ inside oriented graphs from $\mathcal{G}$ with index vector $(4,1,1)$ and $(3,3,0)$, respectively. We identify the parts of $D_2$ as $\{a_1, a_2\},\{b_1, b_2\},\{c_1, c_2\}$. The red edges correspond to reversed edges, and blue edges follow the direction from $V_1$ to $V_2$, $V_2$ to $V_3$, and $V_3$ to $V_1$. We omit edges inside parts for a cleaner picture.
  • Figure 3: A regular tournament on 7 vertices with no copy of $C_6^2$.

Theorems & Definitions (61)

  • Definition 1.2: Turánable
  • Definition 1.3: Tileable
  • Conjecture 1.5: Conjecture 8.3 in DHLMPT
  • Theorem 1.6
  • Theorem 1.7: Bollobás, Häggkvist bollobas
  • Conjecture 1.8
  • Theorem 1.10
  • Proposition 1.12
  • Theorem 1.13
  • Proposition 2.1
  • ...and 51 more