Oriented cycles in digraphs of large outdegree

TL;DR

This paper proves a conjecture that large outdegree in a digraph guarantees the presence of subdivisions of all orientations of cycles of a given length, advancing understanding of cycle structures in directed graphs.

Contribution

It confirms a conjecture that high outdegree ensures subdivisions of all cycle orientations, resolving open questions in directed graph theory.

Findings

Proved the conjecture for all cycle lengths and orientations.

Established minimum outdegree thresholds for cycle subdivisions.

Extended previous results on directed cycle structures.

Abstract

In 1985, Mader conjectured that for every acyclic digraph $F$ there exists $K=K(F)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of $F$. This conjecture remains widely open, even for digraphs $F$ on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomass\'{e} studied special cases of Mader's problem and made the following conjecture: for every $\ell \geq 2$ there exists $K = K(\ell)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of every orientation of a cycle of length $\ell$. We prove this conjecture and answer further open questions raised by Aboulker et al.