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Embedding loose trees in $k$-uniform hypergraphs

Yaobin Chen, Allan Lo

TL;DR

This work advances the theory of spanning tree embeddings to loose trees in $k$-uniform hypergraphs by proving that for every $k\ge 4$, an $n$-vertex hypergraph with minimum $(k-2)$-degree at least $(1/2+\gamma)\binom{n}{k-2}$ contains every spanning loose tree of bounded degree, for large $n$. The authors develop a robust framework based on absorption, weak hypergraph regularity, and a robust reduced-graph structure that supports fractional matchings, reachability, and rotatability. A key innovation is handling the $(k,k-2)$-threshold regime where a tight Hamilton cycle in the reduced graph cannot be guaranteed, achieved via a careful construction of a robust spanning subgraph $G^*$ and a sequence of embedding steps (absorption, almost spanning embedding, and final absorption). The results establish that the loose-tree embedding threshold matches the corresponding perfect-matching threshold for $$k\ge 4$$, thereby resolving a central case and extending prior works on the $k=3$ setting. The findings have significant implications for spanning structures in hypergraphs and illuminate a robust embedding paradigm applicable to broader hypergraph embedding problems.

Abstract

A classical result of Komlós, Sárközy and Szemerédi shows that every large $n$-vertex graph with minimum degree at least $(1/2+γ)n$ contains all spanning trees of bounded degree. We generalised this result to loose spanning hypertrees in $k$-uniform hypergraphs, that is, linear hypergraphs obtained by subsequently adding edges sharing a single vertex with a previous edge. We give a general sufficient condition for embedding loose trees with bounded degree. In particular, we show that for all $k\ge 4$, every $n$-vertex $k$-uniform hypergraph with $n\ge n_0(k,γ, Δ)$ and minimum $(k-2)$-degree at least $(1/2+γ)\binom{n}{k-2}$ contains every spanning loose tree with maximum vertex degree at most $Δ$. This bound is asymptotically tight. This generalises a result of Pehova and Petrova, who proved the case when $k=3$ and of Pavez-Signé, Sanhueza-Matamala and Stein, who considered the codegree threshold for bounded degree tight trees.

Embedding loose trees in $k$-uniform hypergraphs

TL;DR

This work advances the theory of spanning tree embeddings to loose trees in -uniform hypergraphs by proving that for every , an -vertex hypergraph with minimum -degree at least contains every spanning loose tree of bounded degree, for large . The authors develop a robust framework based on absorption, weak hypergraph regularity, and a robust reduced-graph structure that supports fractional matchings, reachability, and rotatability. A key innovation is handling the -threshold regime where a tight Hamilton cycle in the reduced graph cannot be guaranteed, achieved via a careful construction of a robust spanning subgraph and a sequence of embedding steps (absorption, almost spanning embedding, and final absorption). The results establish that the loose-tree embedding threshold matches the corresponding perfect-matching threshold for , thereby resolving a central case and extending prior works on the setting. The findings have significant implications for spanning structures in hypergraphs and illuminate a robust embedding paradigm applicable to broader hypergraph embedding problems.

Abstract

A classical result of Komlós, Sárközy and Szemerédi shows that every large -vertex graph with minimum degree at least contains all spanning trees of bounded degree. We generalised this result to loose spanning hypertrees in -uniform hypergraphs, that is, linear hypergraphs obtained by subsequently adding edges sharing a single vertex with a previous edge. We give a general sufficient condition for embedding loose trees with bounded degree. In particular, we show that for all , every -vertex -uniform hypergraph with and minimum -degree at least contains every spanning loose tree with maximum vertex degree at most . This bound is asymptotically tight. This generalises a result of Pehova and Petrova, who proved the case when and of Pavez-Signé, Sanhueza-Matamala and Stein, who considered the codegree threshold for bounded degree tight trees.

Paper Structure

This paper contains 29 sections, 51 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Structure of loose trees
  5. $\alpha$-perturbed graphs
  6. Proof sketch of Theorem \ref{['Theorem Matching threshold']}
  7. Verifying robustness
  8. A sufficient condition for almost spanning trees embedding
  9. Weak hypergraph regularity lemma
  10. From robust framework to embedding: proof of Theorem \ref{['generaltheorem']}
  11. Absorption lemma
  12. Embedding for almost spanning tree
  13. Complete embedding
  14. Robust spanning subgraph construction
  15. Robust fractional matching
  16. ...and 14 more sections

Key Result

Theorem 1.1

For all $k\ge 2, \gamma>0$ and $\Delta\in \mathbb{N}$, there exists an $n_0$ such that any $k$-graph $G$ on $n\ge n_0$ vertices with $\overline{\delta}_{k-1}(G)\ge 1/2+\gamma$ contains every $n$-vertex $(k-1)$-hypertree $T$ with $\Delta_1(T)\le \Delta$ and the bound is asymptotic tight. In particula

Figures (1)

  • Figure 1: The binary $3$-loose tree $T_{3,2}$.

Theorems & Definitions (123)

  • Theorem 1.1: Pavez-Signé, Sanhueza-Matamala and Stein pavez2024dirac
  • Theorem 1.2: Pehova and Petrova pehova2024embedding
  • Conjecture 1.3: Pehova and Petrova pehova2024embedding
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Definition 2.3: $\alpha$-perturbed degree
  • Proposition 2.4
  • proof
  • Lemma 2.5: lang2022minimum
  • ...and 113 more