Unbounded degree spanning hypertrees in Dirac hypergraphs

TL;DR

This paper extends a classical graph spanning tree result to hypergraphs, establishing optimal degree conditions for the existence of spanning loose hypertrees, and confirms a conjecture for certain hypergraph parameters without using Szemerédi's regularity lemma.

Contribution

It provides asymptotically optimal degree thresholds for spanning loose hypertrees in hypergraphs, generalizing and improving previous results and confirming a conjecture.

Findings

Determined asymptotically optimal degree conditions for loose hypertrees.

Confirmed a conjecture on degree thresholds for hypergraphs.

Avoided using Szemerédi's regularity lemma in the proof.

Abstract

In 2001, Koml\'os, S\'ark\"ozy, and Szemer\'edi proved that every sufficiently large $n$-vertex graph with minimum degree at least $\left(1/2+\gamma\right)n$ contains all spanning trees with maximum degree at most $cn/\log n$. We extend this result to hypergraphs by considering loose hypertrees, which are linear hypergraphs obtained by successively adding edges that share exactly one vertex with a previous edge. For all $k > \ell \geq 2$, we determine asymptotically optimal $\ell$-degree conditions that ensure the existence of all rooted spanning loose hypertrees, without any degree condition, in terms of the $(\ell-1)$-degree threshold for the existence of a perfect matching in $(k-1)$-graphs. As a corollary, we also asymptotically determine the $\ell$-degree threshold for the existence of bounded degree spanning loose hypertrees in $k$-graphs for $k/2 < \ell < k$, confirming a conjecture of Pehova and Petrova in this range. In our proof, we avoid the use of Szemer\'edi's regularity lemma.