Vertex degree sums for perfect matchings in 3-uniform hypergraphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911750375342080 |
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| author | Wang, Yan Zhang, Yi |
| author_facet | Wang, Yan Zhang, Yi |
| contents | Let $n \equiv 0\, (\, \text{mod } 3\,)$ and $H_{n, n/3}^2$ be the 3-graph of order $n$, whose vertex set is partitioned into two sets $S$ and $T$ of size $\frac{1}{3}n+1$ and $\frac{2}{3}n -1$, respectively, and whose edge set consists of all triples with at least $2$ vertices in $T$. Suppose that $n$ is sufficiently large and $H$ is a 3-uniform hypergraph of order $n$ with no isolated vertex. Zhang and Lu [Discrete Math. 341 (2018), 748--758] conjectured that if $deg(u)+deg(v) > 2(\binom{n-1}{2}-\binom{2n/3}{2})$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching or $H$ is a subgraph of $H_{n,n/3}^2$. We construct a counter-example to the conjecture. Furthermore, for all $γ>0$ and let $n \in 3 \mathbb{Z}$ be sufficiently large, we prove that if $deg(u)+deg(v) > (3/5+γ)n^2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching or $H$ is a subgraph of $H_{n,n/3}^2$. This implies a result of Zhang, Zhao and Lu [Electron. J. Combin. 25 (3), 2018]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_03713 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vertex degree sums for perfect matchings in 3-uniform hypergraphs Wang, Yan Zhang, Yi Combinatorics Let $n \equiv 0\, (\, \text{mod } 3\,)$ and $H_{n, n/3}^2$ be the 3-graph of order $n$, whose vertex set is partitioned into two sets $S$ and $T$ of size $\frac{1}{3}n+1$ and $\frac{2}{3}n -1$, respectively, and whose edge set consists of all triples with at least $2$ vertices in $T$. Suppose that $n$ is sufficiently large and $H$ is a 3-uniform hypergraph of order $n$ with no isolated vertex. Zhang and Lu [Discrete Math. 341 (2018), 748--758] conjectured that if $deg(u)+deg(v) > 2(\binom{n-1}{2}-\binom{2n/3}{2})$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching or $H$ is a subgraph of $H_{n,n/3}^2$. We construct a counter-example to the conjecture. Furthermore, for all $γ>0$ and let $n \in 3 \mathbb{Z}$ be sufficiently large, we prove that if $deg(u)+deg(v) > (3/5+γ)n^2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching or $H$ is a subgraph of $H_{n,n/3}^2$. This implies a result of Zhang, Zhao and Lu [Electron. J. Combin. 25 (3), 2018]. |
| title | Vertex degree sums for perfect matchings in 3-uniform hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.03713 |