Embedding loose spanning trees in 3-uniform hypergraphs

TL;DR

This paper extends a classical graph theory result to 3-uniform hypergraphs, proving that certain minimum degree conditions guarantee the embedding of all loose spanning hypertrees with bounded degree.

Contribution

It generalizes the spanning tree embedding theorem from graphs to 3-uniform hypergraphs, establishing tight minimum degree thresholds for loose hypertrees.

Findings

Minimum vertex degree threshold of (5/9 + γ) * binom(n, 2) for embedding loose spanning hypertrees

The bound is asymptotically tight due to the existence of loose trees with perfect matchings

The result applies to large hypergraphs with bounded degree trees

Abstract

In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all $\gamma$ and $\Delta$, and $n$ large, every $n$-vertex 3-uniform hypergraph of minimum vertex degree $(5/9 + \gamma)\binom{n}{2}$ contains every loose spanning tree $T$ with maximum vertex degree $\Delta$. This bound is asymptotically tight, since some loose trees contain perfect matchings.