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 $(1/2+\gamma )n$ 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 γ and Δ, and n large, every n-vertex 3-uniform hypergraph of minimum vertex degree $(5/9+\gamma )\left(\begin{array}{c}n\\ 2\end{array}\right)$ contains every loose spanning tree T with maximum vertex degree Δ. This bound is asymptotically tight, since some loose trees contain perfect matchings.