Matching stability for 3-partite 3-uniform hypergraphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917819163082752 |
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| author | Lu, Hongliang Ma, Xinxin |
| author_facet | Lu, Hongliang Ma, Xinxin |
| contents | Let $n,k,s$ be three integers such that $k\geq 2$ and $n\geq s\geq 1$. Let $H$ be a $k$-partite $k$-uniform hypergraph with $n$ vertices in each class. Aharoni (2017) showed that if $e(H)>(s-1)n^{k-1}$, then $H$ has a matching of size $s$.
In this paper, we give a stability result for 3-partite 3-uniform hypergraphs: if $G$ is a $3$-partite $3$-uniform hypergraph with $n\geq 162$ vertices in each class, $e(G)\geq (s-1)n^2+3n-s$ and $G$ contains no matching of size $s+1$, then $G$ has a vertex cover of size $s$. Our bound is also tight. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_15673 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matching stability for 3-partite 3-uniform hypergraphs Lu, Hongliang Ma, Xinxin Combinatorics Let $n,k,s$ be three integers such that $k\geq 2$ and $n\geq s\geq 1$. Let $H$ be a $k$-partite $k$-uniform hypergraph with $n$ vertices in each class. Aharoni (2017) showed that if $e(H)>(s-1)n^{k-1}$, then $H$ has a matching of size $s$. In this paper, we give a stability result for 3-partite 3-uniform hypergraphs: if $G$ is a $3$-partite $3$-uniform hypergraph with $n\geq 162$ vertices in each class, $e(G)\geq (s-1)n^2+3n-s$ and $G$ contains no matching of size $s+1$, then $G$ has a vertex cover of size $s$. Our bound is also tight. |
| title | Matching stability for 3-partite 3-uniform hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.15673 |