Embedding loose trees in $k$-uniform hypergraphs

TL;DR

This paper extends classical graph embedding results to hypergraphs, establishing conditions under which large uniform hypergraphs contain all spanning loose hypertrees with bounded degree, generalizing prior work for specific cases.

Contribution

It provides a general sufficient condition for embedding loose trees with bounded degree in k-uniform hypergraphs, extending known results to higher uniformities and tight bounds.

Findings

Embedding conditions depend on minimum (k-2)-degree thresholds.

Results are asymptotically tight for all k ≥ 4.

Generalizes previous results for 3-uniform hypergraphs.

Abstract

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi shows that every large $n$-vertex graph with minimum degree at least $(1/2+\gamma)n$ 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 $n\ge n_0(k,\gamma, \Delta)$ and minimum $(k-2)$-degree at least $(1/2+\gamma)\binom{n}{k-2}$ contains every spanning loose tree with maximum vertex degree at most $\Delta$. This bound is asymptotically tight. This generalises a result of Pehova and Petrova, who proved the case when $k=3$ and of Pavez-Sign\'e, Sanhueza-Matamala and Stein, who considered the codegree threshold for bounded degree tight trees.