Dipaths of length at least double the minimum outdegree

TL;DR

This paper investigates a conjecture related to the existence of long directed paths in oriented graphs with minimum outdegree k, providing bounds and reductions to understand potential counterexamples.

Contribution

It introduces reductions and bounds on connectivity and size of counterexamples for Thomassé's conjecture, and proposes a new related conjecture.

Findings

Established bounds on connectivity and vertices for potential counterexamples

Presented reductions simplifying the problem

Proposed a new conjecture about vertices belonging to long dipaths

Abstract

A special case of a conjecture by Thomass\'e is that any oriented graph with minimum outdegree k contains a dipath of length 2k. For the sake of proving whether or not a counterexample exists, we present reductions and establish bounds on both connectivity and the number of vertices in a counterexample. We further conjecture that in any oriented graph with minimum outdegree k, every vertex belongs to a dipath of length 2k.