Perfect matchings in random sparsifications of Dirac hypergraphs
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910411347984384 |
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| author | Kang, Dong Yeap Kelly, Tom Kühn, Daniela Osthus, Deryk Pfenninger, Vincent |
| author_facet | Kang, Dong Yeap Kelly, Tom Kühn, Daniela Osthus, Deryk Pfenninger, Vincent |
| contents | For all integers $n \geq k > d \geq 1$, let $m_{d}(k,n)$ be the minimum integer $D \geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal H$ with minimum $d$-degree $δ_{d}(\mathcal H)$ at least $D$ has an optimal matching. For every fixed integer $k \geq 3$, we show that for $n \in k \mathbb{N}$ and $p = Ω(n^{-k+1} \log n)$, if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $δ_{k-1}(\mathcal H) \geq m_{k-1}(k,n)$, then a.a.s.\ its $p$-random subhypergraph $\mathcal H_p$ contains a perfect matching. Moreover, for every fixed integer $d < k$ and $γ> 0$, we show that the same conclusion holds if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $δ_d(\mathcal H) \geq m_{d}(k,n) + γ\binom{n - d}{k - d}$. Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as ``robust'' versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, $\mathcal H$ has at least $\exp((1-1/k)n \log n - Θ(n))$ many perfect matchings, which is best possible up to an $\exp(Θ(n))$ factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01325 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Perfect matchings in random sparsifications of Dirac hypergraphs Kang, Dong Yeap Kelly, Tom Kühn, Daniela Osthus, Deryk Pfenninger, Vincent Combinatorics For all integers $n \geq k > d \geq 1$, let $m_{d}(k,n)$ be the minimum integer $D \geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal H$ with minimum $d$-degree $δ_{d}(\mathcal H)$ at least $D$ has an optimal matching. For every fixed integer $k \geq 3$, we show that for $n \in k \mathbb{N}$ and $p = Ω(n^{-k+1} \log n)$, if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $δ_{k-1}(\mathcal H) \geq m_{k-1}(k,n)$, then a.a.s.\ its $p$-random subhypergraph $\mathcal H_p$ contains a perfect matching. Moreover, for every fixed integer $d < k$ and $γ> 0$, we show that the same conclusion holds if $\mathcal H$ is an $n$-vertex $k$-uniform hypergraph with $δ_d(\mathcal H) \geq m_{d}(k,n) + γ\binom{n - d}{k - d}$. Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as ``robust'' versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, $\mathcal H$ has at least $\exp((1-1/k)n \log n - Θ(n))$ many perfect matchings, which is best possible up to an $\exp(Θ(n))$ factor. |
| title | Perfect matchings in random sparsifications of Dirac hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2211.01325 |