Unbounded degree spanning hypertrees in Dirac hypergraphs

Abstract

In 2001, Komlós, Sárközy, and Szemerédi proved that every sufficiently large $n$-vertex graph with minimum degree at least $\left(1/2+{\hat{I}}^3\right)n$ 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 $k>\ell \ge 2$, 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 Szemerédi's regularity lemma.

15 pages