Vertex degree sums for rainbow matchings in 3-uniform hypergraphs
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908276617117696 |
|---|---|
| author | Liu, Haorui Lu, Mei Wang, Yan Zhang, Yi |
| author_facet | Liu, Haorui Lu, Mei Wang, Yan Zhang, Yi |
| contents | Let $n \in 3\mathbb{Z}$ be sufficiently large. Zhang, Zhao and Lu proved that if $H$ is a 3-uniform hypergraph with $n$ vertices and no isolated vertices, and if $deg(u)+deg(v) > \frac{2}{3}n^2 - \frac{8}{3}n + 2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $ H $ admits a perfect matching. In this paper, we prove that the rainbow version of Zhang, Zhao and Lu's result is asymptotically true. More specifically, let $δ> 0$ and $ F_1, F_2, \dots, F_{n/3} $ be 3-uniform hypergraphs on a common set of $n$ vertices. For each $ i \in [n/3] $, suppose that $F_i$ has no isolated vertices and $deg_{F_i}(u)+deg_{F_i}(v) > \left( \frac{2}{3} + δ\right)n^2$ holds for any two vertices $u$ and $v$ that are contained in some edge of $F_i$. Then $ \{ F_1, F_2, \dots, F_{n/3} \} $ admits a rainbow matching. Note that this result is asymptotically tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14968 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vertex degree sums for rainbow matchings in 3-uniform hypergraphs Liu, Haorui Lu, Mei Wang, Yan Zhang, Yi Combinatorics Let $n \in 3\mathbb{Z}$ be sufficiently large. Zhang, Zhao and Lu proved that if $H$ is a 3-uniform hypergraph with $n$ vertices and no isolated vertices, and if $deg(u)+deg(v) > \frac{2}{3}n^2 - \frac{8}{3}n + 2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $ H $ admits a perfect matching. In this paper, we prove that the rainbow version of Zhang, Zhao and Lu's result is asymptotically true. More specifically, let $δ> 0$ and $ F_1, F_2, \dots, F_{n/3} $ be 3-uniform hypergraphs on a common set of $n$ vertices. For each $ i \in [n/3] $, suppose that $F_i$ has no isolated vertices and $deg_{F_i}(u)+deg_{F_i}(v) > \left( \frac{2}{3} + δ\right)n^2$ holds for any two vertices $u$ and $v$ that are contained in some edge of $F_i$. Then $ \{ F_1, F_2, \dots, F_{n/3} \} $ admits a rainbow matching. Note that this result is asymptotically tight. |
| title | Vertex degree sums for rainbow matchings in 3-uniform hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.14968 |