Rainbow planar Tur{á}n numbers of cycles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915601927110656 |
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| author | Liu, Xiaonan |
| author_facet | Liu, Xiaonan |
| contents | The rainbow Tur{á}n number of a fixed graph $H$, denoted by ${\text{ex}}^*(n,H)$, is the maximum number of edges in an $n$-vertex graph such that it admits a proper edge coloring with no rainbow $H$. We study this problem in planar setting. The rainbow planar Tur{á}n number of a graph $H$, denoted by ${\text{ex}_{\mathcal{P}}}^*(n,H)$, is the maximum number of edges in an $n$-vertex planar graph such that it has a proper edge coloring with no rainbow $H$. We consider the rainbow planar Tur{á}n number of cycles. Since $C_3$ is complete, ${\text{ex}_{\mathcal{P}}}^*(n, C_3)$ is exactly its planar Tur{á}n number, which is $2n-4$ for $n\ge 3$. We show that ${\text{ex}_{\mathcal{P}}}^*(n, C_4)=3n-6$ for $n=k^2-3k+2$ where $k\ge 5$, and ${\text{ex}_{\mathcal{P}}}^*(n,C_k)=3n-6$ for all $k\ge 5$ and $n\ge 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rainbow planar Tur{á}n numbers of cycles Liu, Xiaonan Combinatorics 05C10, 05C15, 05C35, 05C38 The rainbow Tur{á}n number of a fixed graph $H$, denoted by ${\text{ex}}^*(n,H)$, is the maximum number of edges in an $n$-vertex graph such that it admits a proper edge coloring with no rainbow $H$. We study this problem in planar setting. The rainbow planar Tur{á}n number of a graph $H$, denoted by ${\text{ex}_{\mathcal{P}}}^*(n,H)$, is the maximum number of edges in an $n$-vertex planar graph such that it has a proper edge coloring with no rainbow $H$. We consider the rainbow planar Tur{á}n number of cycles. Since $C_3$ is complete, ${\text{ex}_{\mathcal{P}}}^*(n, C_3)$ is exactly its planar Tur{á}n number, which is $2n-4$ for $n\ge 3$. We show that ${\text{ex}_{\mathcal{P}}}^*(n, C_4)=3n-6$ for $n=k^2-3k+2$ where $k\ge 5$, and ${\text{ex}_{\mathcal{P}}}^*(n,C_k)=3n-6$ for all $k\ge 5$ and $n\ge 3$. |
| title | Rainbow planar Tur{á}n numbers of cycles |
| topic | Combinatorics 05C10, 05C15, 05C35, 05C38 |
| url | https://arxiv.org/abs/2511.04066 |