Anti-Ramsey Number of Stars in 3-uniform hypergraphs
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917137146183680 |
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| author | Lu, Hongliang Luo, Xinyue Ma, Xinxin |
| author_facet | Lu, Hongliang Luo, Xinyue Ma, Xinxin |
| contents | An edge-colored hypergraph is called \emph{a rainbow hypergraph} if all the colors on its edges are distinct. Given two positive integers $n,r$ and an $r$-uniform hypergraph $\mathcal{G}$, the anti-Ramsey number $ar_r(n,\mathcal{G})$ is defined to be the minimum number of colors $t$ such that there exists a rainbow copy of $\mathcal{G}$ in any exactly $t$-edge-coloring of the complete $r$-uniform hypergraph of order $n$. Let $ \mathcal{F}_k $ denote the 3-graph ($k$-star) consisting of $k$ edges sharing exactly one vertex. Tang, Li and Yan \cite{YTG} determined the value of $ar_3(n,\mathcal{F}_3)$ when $n\geq 20$. In this paper, we determine the anti-Ramsey number $ar_3(n,\mathcal{F}_{k+1})$, where $k\geq 3$ and $n> \frac{5}{2}k^3+\frac{15}{2}k^2+26k-3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anti-Ramsey Number of Stars in 3-uniform hypergraphs Lu, Hongliang Luo, Xinyue Ma, Xinxin Combinatorics An edge-colored hypergraph is called \emph{a rainbow hypergraph} if all the colors on its edges are distinct. Given two positive integers $n,r$ and an $r$-uniform hypergraph $\mathcal{G}$, the anti-Ramsey number $ar_r(n,\mathcal{G})$ is defined to be the minimum number of colors $t$ such that there exists a rainbow copy of $\mathcal{G}$ in any exactly $t$-edge-coloring of the complete $r$-uniform hypergraph of order $n$. Let $ \mathcal{F}_k $ denote the 3-graph ($k$-star) consisting of $k$ edges sharing exactly one vertex. Tang, Li and Yan \cite{YTG} determined the value of $ar_3(n,\mathcal{F}_3)$ when $n\geq 20$. In this paper, we determine the anti-Ramsey number $ar_3(n,\mathcal{F}_{k+1})$, where $k\geq 3$ and $n> \frac{5}{2}k^3+\frac{15}{2}k^2+26k-3$. |
| title | Anti-Ramsey Number of Stars in 3-uniform hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.09747 |