Dirac's theorem for linear hypergraphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908284559032320 |
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| author | Im, Seonghyuk Lee, Hyunwoo |
| author_facet | Im, Seonghyuk Lee, Hyunwoo |
| contents | Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $δ(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex $k$-uniform linear hypergraph $H$ with minimum degree at least $\frac{n}{k} + Ω(1)$ contains a matching that covers at least $(1-o(1))n$ vertices. This minimum degree condition is asymptotically tight and obtaining a perfect matching is impossible with any degree condition. Furthermore, we show that if $δ(H) \geq (\frac{1}{k}+o(1))n$, then $H$ contains almost spanning linear cycles, almost spanning hypertrees with $o(n)$ leaves, and ``long subdivisions'' of any $o(\sqrt{n})$-vertex graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14269 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirac's theorem for linear hypergraphs Im, Seonghyuk Lee, Hyunwoo Combinatorics Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $δ(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex $k$-uniform linear hypergraph $H$ with minimum degree at least $\frac{n}{k} + Ω(1)$ contains a matching that covers at least $(1-o(1))n$ vertices. This minimum degree condition is asymptotically tight and obtaining a perfect matching is impossible with any degree condition. Furthermore, we show that if $δ(H) \geq (\frac{1}{k}+o(1))n$, then $H$ contains almost spanning linear cycles, almost spanning hypertrees with $o(n)$ leaves, and ``long subdivisions'' of any $o(\sqrt{n})$-vertex graphs. |
| title | Dirac's theorem for linear hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.14269 |