Antipaths in oriented graphs

TL;DR

This paper proves that oriented graphs with sufficiently high minimum semidegree contain antidirected paths of any given length, confirming a weakened version of a conjecture related to antidirected paths.

Contribution

It establishes a minimum semidegree condition guaranteeing the existence of antidirected paths of arbitrary length in oriented graphs, advancing understanding of graph orientation properties.

Findings

Minimum semidegree at least (3k-2)/4 ensures antidirected paths of length k

A weaker semidegree sequence condition also suffices

Confirms a weakened form of a conjecture on antidirected paths

Abstract

We show that for any natural number $k \ge 1$, any oriented graph $D$ of minimum semidegree at least $(3k- 2)/4$ contains an antidirected path of length $k$. In fact, a slightly weaker condition on the semidegree sequence of $D$ suffices, and as a consequence, we confirm a weakened antidirected path version of a conjecture of Addario-Berry, Havet, Linhares Sales, Thomass\'e and Reed.