Embedding loose trees in $k$-uniform hypergraphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916602899857408 |
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| author | Chen, Yaobin Lo, Allan |
| author_facet | Chen, Yaobin Lo, Allan |
| contents | A classical result of Komlós, Sárközy and Szemerédi shows that every large $n$-vertex graph with minimum degree at least $(1/2+γ)n$ contains all spanning trees of bounded degree. We generalised this result to loose spanning hypertrees in $k$-uniform hypergraphs, that is, linear hypergraphs obtained by subsequently adding edges sharing a single vertex with a previous edge.
We give a general sufficient condition for embedding loose trees with bounded degree. In particular, we show that for all $k\ge 4$, every $n$-vertex $k$-uniform hypergraph with $n\ge n_0(k,γ, Δ)$ and minimum $(k-2)$-degree at least $(1/2+γ)\binom{n}{k-2}$ contains every spanning loose tree with maximum vertex degree at most $Δ$. This bound is asymptotically tight. This generalises a result of Pehova and Petrova, who proved the case when $k=3$ and of Pavez-Signé, Sanhueza-Matamala and Stein, who considered the codegree threshold for bounded degree tight trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Embedding loose trees in $k$-uniform hypergraphs Chen, Yaobin Lo, Allan Combinatorics A classical result of Komlós, Sárközy and Szemerédi shows that every large $n$-vertex graph with minimum degree at least $(1/2+γ)n$ contains all spanning trees of bounded degree. We generalised this result to loose spanning hypertrees in $k$-uniform hypergraphs, that is, linear hypergraphs obtained by subsequently adding edges sharing a single vertex with a previous edge. We give a general sufficient condition for embedding loose trees with bounded degree. In particular, we show that for all $k\ge 4$, every $n$-vertex $k$-uniform hypergraph with $n\ge n_0(k,γ, Δ)$ and minimum $(k-2)$-degree at least $(1/2+γ)\binom{n}{k-2}$ contains every spanning loose tree with maximum vertex degree at most $Δ$. This bound is asymptotically tight. This generalises a result of Pehova and Petrova, who proved the case when $k=3$ and of Pavez-Signé, Sanhueza-Matamala and Stein, who considered the codegree threshold for bounded degree tight trees. |
| title | Embedding loose trees in $k$-uniform hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.04783 |