Dirac-type conditions for spanning bounded-degree hypertrees

TL;DR

This paper extends classical graph spanning tree results to hypergraphs, proving that high minimum codegree guarantees the presence of all bounded-degree spanning tight hypertrees, with extensions to quasirandom hypergraphs.

Contribution

It generalizes a fundamental spanning tree theorem from graphs to hypergraphs, establishing near-optimal codegree conditions for hypergraph containment of bounded-degree spanning tight hypertrees.

Findings

High minimum codegree ensures spanning tight hypertrees in hypergraphs.

The results are asymptotically sharp, matching known bounds.

Extensions to quasirandom hypergraphs are also established.

Abstract

We prove that for fixed $k$, every $k$-uniform hypergraph on $n$ vertices and of minimum codegree at least $n/2+o(n)$ contains every spanning tight $k$-tree of bounded vertex degree as a sub\-graph. This generalises a well-known result of Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions.