# Typical Structure of Oriented Graphs and Digraphs with Forbidden Blow-Up Transitive Triangles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notations, Tools and Results

**Theorem**

**1**

**.**Let H be an oriented graph with $h:=v\left(H\right)$ and $e\left(H\right)\ge 2$, and let $a\in \mathbb{R}$ with $a\ge 1$. For every $\u03f5>0$, there exists $c>0$ such that for all sufficiently large N, there exists a collection $\mathcal{C}$ of digraphs on the vertex set $\left[n\right]$ with the following properties.

- (a)
- For every H-free digraph I on $\left[N\right]$ there exists $G\in \mathcal{C}$ such that $I\subset G$.
- (b)
- Every digraph $G\in \mathcal{C}$ contains at most $\u03f5{N}^{h}$ copies of H, and ${e}_{a}\left(G\right)\le e{x}_{a}(N,H)+\u03f5{N}^{2}$.
- (c)
- $log\left|\mathcal{C}\right|\le c{N}^{2-1/m\left(H\right)}logN$.

**Lemma**

**1**

**.**Let $a\in (\frac{3}{2},2]$ be a real number and let $r,n\in \mathbb{N}$. Then $e{x}_{a}(n,{T}_{r+1})=a\xb7{t}_{r}\left(n\right)$, and $D{T}_{r}\left(n\right)$ is the unique extremal ${T}_{r+1}$-free digraph on n vertices.

**Theorem**

**2.**

**Example**

**1.**

**Theorem**

**3.**

- (i)
- All but at most $f(n,{T}_{3}^{t}){2}^{-\u03f5{n}^{2}}$${T}_{3}^{t}$-free oriented graphs on n vertices can be made bipartite by changing at most $\alpha {n}^{2}$ edges.
- (ii)
- All but at most ${f}^{*}(n,{T}_{3}^{t}){2}^{-\u03f5{n}^{2}}$${T}_{3}^{t}$-free digraphs on n vertices can be made bipartite by changing at most $\alpha {n}^{2}$ edges.

## 3. The Regularity Lemma and the Proof of Theorem 2

**Lemma**

**2**

**.**For every $\u03f5\in (0,1)$ and all integers ${M}^{\prime},{M}^{\u2033}$ there are integers M and ${n}_{0}$ such that if

- G is a digraph on $n\ge {n}_{0}$ vertices,
- ${U}_{0},\dots ,{U}_{{M}^{\u2033}}$ is a partition of the vertex set of G,
- $d\in [0,1]$ is any real number,

- (1)
- ${M}^{\prime}\le k\le M$,
- (2)
- $|{V}_{0}|\le \u03f5\xb7n$,
- (3)
- $|{V}_{1}|=\dots =|{V}_{k}|=\ell $,
- (4)
- ${V}_{0},\dots ,{V}_{k}$ refines the partition ${U}_{0},\dots ,{U}_{{M}^{\u2033}}$,
- (5)
- ${d}_{{G}^{\prime}}^{+}\left(x\right)>{d}_{G}^{+}\left(x\right)-(d+\u03f5)n$ for all vertices x of G,
- (6)
- ${d}_{{G}^{\prime}}^{-}\left(x\right)>{d}_{G}^{-}\left(x\right)-(d+\u03f5)n$ for all vertices x of G,
- (7)
- ${G}^{\prime}\left[{V}_{i}\right]$ is empty for all $i=1,\dots ,k$,
- (8)
- the bipartite oriented graph ${({V}_{i},{V}_{j})}_{{G}^{\prime}}$ is ϵ-regular and has density either 0 or density at least d for all $1\le i,j\le k$ and $i\ne j$.

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

- Case (i):
- ${u}_{i}$ are both out-neighbor and in-neighbor of ${u}_{j}$ in H, we delete all those vertices from ${Y}_{i}$ that are not adjacent to ${v}_{j}$ with double edges.
- Case (ii):
- ${u}_{i}$ is just the out-neighbor of ${u}_{j}$ in H, we delete all those vertices from ${Y}_{i}$ that are not the out-neighbor of ${v}_{j}$.
- Case (iii):
- ${u}_{i}$ is just the in-neighbour of ${u}_{j}$ in H, we delete all those vertices from ${Y}_{i}$ that are not the in-neighbour of ${v}_{j}$.

**Proof of Theorem**

**2.**

**Corollary**

**1.**

**Proof of Corollary**

**1.**

## 4. Stability of Digraphs and Proof of Theorem 3

**Lemma**

**5.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**of**

**Stability.**

**Lemma**

**6.**

**Proof of Theorem**

**3.**

## 5. Concluding Remarks

**Conjecture**

**2.**

- (i)
- Almost all ${T}_{r+1}^{t}$-free oriented graph are r-partite.
- (ii)
- Almost all ${T}_{r+1}^{t}$-free digraph are r-partite.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Liang, M.; Liu, J.
Typical Structure of Oriented Graphs and Digraphs with Forbidden Blow-Up Transitive Triangles. *Symmetry* **2022**, *14*, 2551.
https://doi.org/10.3390/sym14122551

**AMA Style**

Liang M, Liu J.
Typical Structure of Oriented Graphs and Digraphs with Forbidden Blow-Up Transitive Triangles. *Symmetry*. 2022; 14(12):2551.
https://doi.org/10.3390/sym14122551

**Chicago/Turabian Style**

Liang, Meili, and Jianxi Liu.
2022. "Typical Structure of Oriented Graphs and Digraphs with Forbidden Blow-Up Transitive Triangles" *Symmetry* 14, no. 12: 2551.
https://doi.org/10.3390/sym14122551