Antidirected paths in oriented graphs

TL;DR

This paper proves that oriented graphs with sufficiently high minimum semidegree or edge count necessarily contain antidirected paths of a given length, confirming conjectures in the field.

Contribution

It establishes the minimum semidegree and edge threshold conditions for the existence of antidirected paths of length k in oriented graphs, resolving related conjectures.

Findings

Graphs with semidegree above the threshold contain antidirected paths of length k

Graphs with more than a certain number of edges contain antidirected paths of length k

Asymptotic proof of conjectures by Stein and others

Abstract

We show that for any integer $k \ge 4$, every oriented graph with minimum semidegree bigger than $\frac{1}{2}(k-1+\sqrt{k-3})$ contains an antidirected path of length $k$. Consequently, every oriented graph on $n$ vertices with more than $(k-1+\sqrt{k-3})n$ edges contains an antidirected path of length $k$. This asymptotically proves the antidirected path version of a conjecture of Stein and of a conjecture of Addario-Berry, Havet, Linhares Sales, Reed and Thomass\'e, respectively.