New Bounds on the Anti-Ramsey Number of Independent Triangles
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915333103681536 |
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| author | Lu, Hongliang Luo, Xinyue Ma, Xinxin |
| author_facet | Lu, Hongliang Luo, Xinyue Ma, Xinxin |
| contents | An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. Given a positive integer $n$ and a graph $G$, the \textit{anti-Ramsey number} $ar(n,G)$ is defined to be the minimum number of colors $r$ such that there exists a rainbow copy of $G$ in any exactly $r$-edge-coloring of $K_n$.
Wu et al. (Anti-Ramsey numbers for vertex-disjoint triangles, \emph{Discrete. Math.}, \textbf{346} (2022), 113123) determined the anti-Ramsey number $ar(n, kK_3)$ for $n\geq 2k^2-k+2 $. In this paper, we extend this result by improving the lower bound on $n$ to $n\geq 15k+57$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07115 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New Bounds on the Anti-Ramsey Number of Independent Triangles Lu, Hongliang Luo, Xinyue Ma, Xinxin Combinatorics An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. Given a positive integer $n$ and a graph $G$, the \textit{anti-Ramsey number} $ar(n,G)$ is defined to be the minimum number of colors $r$ such that there exists a rainbow copy of $G$ in any exactly $r$-edge-coloring of $K_n$. Wu et al. (Anti-Ramsey numbers for vertex-disjoint triangles, \emph{Discrete. Math.}, \textbf{346} (2022), 113123) determined the anti-Ramsey number $ar(n, kK_3)$ for $n\geq 2k^2-k+2 $. In this paper, we extend this result by improving the lower bound on $n$ to $n\geq 15k+57$. |
| title | New Bounds on the Anti-Ramsey Number of Independent Triangles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.07115 |