Minimum vertex degree conditions for loose spanning trees in 3-graphs

Abstract

In 1995, Koml\‘os, S\‘ark\"ozy and Szemer\‘edi showed that for large $n$, every $n$-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, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge, in 3-graphs. We show that for all $\gamma$ and $\Delta$, and $n$ large, every $n$-vertex 3-uniform hypergraph of minimum vertex degree $(5/9 + \gamma)\binom{n}{2}$ contains every loose spanning tree with maximum vertex degree $\Delta$. This bound is asymptotically tight, since some loose trees contain perfect matchings.