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Almost all $C_k$-free oriented graphs have $Θ(n)$ backwards edges

Jianxi Liu, Meili Liang

Abstract

We prove a conjecture of Kühn, Osthus, Townsend and Zhao \cite{kuhn2017structure} stating that almost every $C_k$-free oriented graph on $n$ vertices has $Θ(n)$ backwards edges in a transitive-optimal ordering. The same holds for $C_k$-free digraphs when $k$ is even. Our proof combines the hypergraph container method with a stability analysis and an inductive counting argument. As a byproduct, we also determine the typical structure of oriented graphs and digraphs that avoid the blow-up $C_{k}^t$, extending the main result of \cite{kuhn2017structure} to the blown-up setting.

Almost all $C_k$-free oriented graphs have $Θ(n)$ backwards edges

Abstract

We prove a conjecture of Kühn, Osthus, Townsend and Zhao \cite{kuhn2017structure} stating that almost every -free oriented graph on vertices has backwards edges in a transitive-optimal ordering. The same holds for -free digraphs when is even. Our proof combines the hypergraph container method with a stability analysis and an inductive counting argument. As a byproduct, we also determine the typical structure of oriented graphs and digraphs that avoid the blow-up , extending the main result of \cite{kuhn2017structure} to the blown-up setting.
Paper Structure (9 sections, 1 theorem, 16 equations)

This paper contains 9 sections, 1 theorem, 16 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Conjecture 1.4(i): oriented graphs
  4. Contradiction setup
  5. Proof of (4) for a fixed ordering
  6. Upper bound
  7. Proof of Conjecture 1.4(ii): digraphs with even $k$
  8. Extension to blow-ups $C_k^t$
  9. Concluding remarks

Key Result

Lemma 2.1

For all sufficiently large $n$ and any integers $0\le m_1\le m_2/(2\log n)$, where $\mathcal{F}_m$ denotes the set of $C_k$-free oriented graphs on $n$ vertices with exactly $m$ backwards edges. In particular, taking $m_1=0$ yields

Theorems & Definitions (1)

  • Lemma 2.1: kuhn2017structure