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arXiv · 2508.06843

Unbounded degree spanning hypertrees in Dirac hypergraphs

Yaobin Chen·Seonghyuk Im·Junchi Zhang
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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.

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Yaobin Chen, Seonghyuk Im, Junchi Zhang. 2025-08-09. Unbounded degree spanning hypertrees in Dirac hypergraphs. https://arxiv.org/abs/2508.06843

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