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Proof of the KAMAK tree conjecture

Micha Christoph, Raphael Steiner

TL;DR

We prove that every grounded forest is $δ^+$-enforcible, i.e., for each grounded forest $F$ there exists a constant $d(F)$ such that any digraph with minimum out-degree at least $d(F)$ contains a subdigraph isomorphic to $F$. The core method introduces $(k,d)$-broom digraphs and a suite of auxiliary lemmas (including pruning, typing, and probabilistic reduction via the Lovász Local Lemma) to embed grounded trees into such brooms, and hence into any digraph of sufficiently large minimum out-degree. A key structural insight is that δ^+-enforcible digraphs are precisely grounded forests, and the main theorem confirms the sufficiency by constructing proper copies of max-grounded trees inside $(k,d)$-brooms. The authors also derive a bound showing $d_k \uparrow 2^{(Ck)^k}$ for the minimum out-degree needed to force all rooted trees on $k$ vertices, and discuss potential strengthening and generalizations.

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, Klimošová, Kucheriya, Mikšaník, 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.

Proof of the KAMAK tree conjecture

TL;DR

We prove that every grounded forest is -enforcible, i.e., for each grounded forest there exists a constant such that any digraph with minimum out-degree at least contains a subdigraph isomorphic to . The core method introduces -broom digraphs and a suite of auxiliary lemmas (including pruning, typing, and probabilistic reduction via the Lovász Local Lemma) to embed grounded trees into such brooms, and hence into any digraph of sufficiently large minimum out-degree. A key structural insight is that δ^+-enforcible digraphs are precisely grounded forests, and the main theorem confirms the sufficiency by constructing proper copies of max-grounded trees inside -brooms. The authors also derive a bound showing for the minimum out-degree needed to force all rooted trees on vertices, and discuss potential strengthening and generalizations.

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, Klimošová, Kucheriya, Mikšaník, 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.

Paper Structure

This paper contains 4 sections, 1 figure.

Table of Contents

  1. Introduction
  2. Auxiliary lemmas
  3. Embedding the tree
  4. Concluding remarks

Figures (1)

  • Figure 1: A grounded tree. The height function is reflected by the heights of vertices in the picture (with height increasing from top to bottom) and the vertices of in-degree at least two are circled.