Anti-Ramsey Number of Friendship Graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915025882447872 |
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| author | Liu, Wenke Lu, Hongliang Luo, Xinyue |
| author_facet | Liu, Wenke Lu, Hongliang Luo, Xinyue |
| contents | An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. For a given positive integer $n$ and a family of graphs $\mathcal{G}$, the anti-Ramsey number $ar(n, \mathcal{G})$ is the smallest number of colors $r$ required to ensure that, no matter how the edges of the complete graph $K_n$ are colored using exactly $r$ colors, there will always be a rainbow copy of some graph $G$ from the family $\mathcal{G}$. A friendship graph $F_k$ is the graph obtained by combining $k$ triangles that share a common vertex. In this paper, we determine the anti-Ramsey number $ar(n, \{F_k\})$ for large values of $n$. Additionally, we also determine the $ar(n, \{K_{1,k}, kK_2\}$, where $K_{1,k}$ is a star graph with $ k+1$ vertices and $kK_2$ is a matching of size $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anti-Ramsey Number of Friendship Graphs Liu, Wenke Lu, Hongliang Luo, Xinyue Combinatorics An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. For a given positive integer $n$ and a family of graphs $\mathcal{G}$, the anti-Ramsey number $ar(n, \mathcal{G})$ is the smallest number of colors $r$ required to ensure that, no matter how the edges of the complete graph $K_n$ are colored using exactly $r$ colors, there will always be a rainbow copy of some graph $G$ from the family $\mathcal{G}$. A friendship graph $F_k$ is the graph obtained by combining $k$ triangles that share a common vertex. In this paper, we determine the anti-Ramsey number $ar(n, \{F_k\})$ for large values of $n$. Additionally, we also determine the $ar(n, \{K_{1,k}, kK_2\}$, where $K_{1,k}$ is a star graph with $ k+1$ vertices and $kK_2$ is a matching of size $k$. |
| title | Anti-Ramsey Number of Friendship Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.08475 |