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arXiv · 2301.09630

Embedding loose spanning trees in 3-uniform hypergraphs

Yanitsa Pehova·Kalina Petrova
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Abstract

In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi 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 $\gamma$ 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 $T$ with maximum vertex degree $\Delta$. This bound is asymptotically tight, since some loose trees contain perfect matchings.

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Yanitsa Pehova, Kalina Petrova. 2023-01-23. Embedding loose spanning trees in 3-uniform hypergraphs. https://arxiv.org/abs/2301.09630

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