Embedding loose spanning trees in 3-uniform hypergraphs

Abstract

In 1995, Komlós, Sárközy and Szemerédi showed that every large $n$-vertex graph with minimum degree at least $(1/2+{\hat{I}}^3)n$ 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 ${\hat{I}}^3$ and $\hat{I}”$, and $n$ large, every $n$-vertex 3-uniform hypergraph of minimum vertex degree $(5/9+{\hat{I}}^3)\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ contains every loose spanning tree $T$ with maximum vertex degree $\hat{I}”$. This bound is asymptotically tight, since some loose trees contain perfect matchings.

23 pages, 3 figures