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

TL;DR

This paper proves that almost all $C_k$-free oriented graphs have a linear number of backwards edges in a transitive-optimal ordering, confirming a conjecture and extending results to blown-up cycles.

Contribution

It confirms a conjecture about the structure of $C_k$-free oriented graphs and digraphs, and characterizes the typical structure of graphs avoiding blow-up cycles.

Findings

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

The same linear bound holds for $C_k$-free digraphs when $k$ is even.

Extended the structural understanding to graphs avoiding blow-up cycles.

Abstract

We prove a conjecture of K\"uhn, Osthus, Townsend and Zhao \cite{kuhn2017structure} stating that almost every $C_k$-free oriented graph on $n$ vertices has $\Theta(n)$ 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.