On the Keevash-Knox-Mycroft Conjecture
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866908916384792576 |
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| author | Gan, Luyining Han, Jie |
| author_facet | Gan, Luyining Han, Jie |
| contents | Given $1\le \ell <k$ and $δ\ge0$, let $\textbf{PM}(k,\ell,δ)$ be the decision problem for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs with minimum $\ell$-degree at least $δ\binom{n-\ell}{k-\ell}$. For $k\ge 3$, $\textbf{PM}(k,\ell,0)$ was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that $\textbf{PM}(k, \ell, δ)$ is in P for every $δ> 1-(1-1/k)^{k-\ell}$ and verified the case $\ell=k-1$.
In this paper we show that this problem can be reduced to the study of the minimum $\ell$-degree condition forcing the existence of fractional perfect matchings. Together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for $\ell\ge 0.4k$. Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_04246 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Keevash-Knox-Mycroft Conjecture Gan, Luyining Han, Jie Combinatorics Computational Complexity Given $1\le \ell <k$ and $δ\ge0$, let $\textbf{PM}(k,\ell,δ)$ be the decision problem for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs with minimum $\ell$-degree at least $δ\binom{n-\ell}{k-\ell}$. For $k\ge 3$, $\textbf{PM}(k,\ell,0)$ was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that $\textbf{PM}(k, \ell, δ)$ is in P for every $δ> 1-(1-1/k)^{k-\ell}$ and verified the case $\ell=k-1$. In this paper we show that this problem can be reduced to the study of the minimum $\ell$-degree condition forcing the existence of fractional perfect matchings. Together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for $\ell\ge 0.4k$. Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists. |
| title | On the Keevash-Knox-Mycroft Conjecture |
| topic | Combinatorics Computational Complexity |
| url | https://arxiv.org/abs/2202.04246 |