Proof of the KAMAK tree conjecture

TL;DR

This paper proves the KAMAK tree conjecture, establishing that certain oriented forests necessarily appear as subgraphs in all large enough digraphs with high minimum out-degree, advancing extremal digraph theory.

Contribution

It confirms the conjecture that all oriented forests with a specific structure appear in any sufficiently large digraph with high minimum out-degree.

Findings

Confirmed the KAMAK tree conjecture.

Characterized the class of oriented forests that must appear.

Extended understanding of extremal properties in digraphs.

Abstract

There are many intriguing questions in extremal graph theory that are well-understood in the undirected setting and yet remain elusive for digraphs. A natural instance of such a problem was recently studied by Hons, Klimo\v{s}ov\'{a}, Kucheriya, Mik\v{s}an\'{i}k, Tkadlec and Tyomkyn: What are the digraphs that have to appear as a subgraph in all digraphs of sufficiently large minimum out-degree? Hons et al. showed that all such digraphs must be oriented forests with a specific structure, and conjectured that vice-versa all oriented forests with this specific structure appear in any digraph of sufficiently large minimum out-degree. In this paper, we confirm their conjecture.