Planar Turán number of disjoint union of $C_3$ and $C_5$
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915404369100800 |
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| author | Li, Luyi Li, Ping Yan, Guiying Zhou, Qiang |
| author_facet | Li, Luyi Li, Ping Yan, Guiying Zhou, Qiang |
| contents | The planar Turán number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Turán number of $k\geq 3$ vertex-disjoint union of cycles is the trivial value $3n-6$. Let $C_{\ell}$ denote the cycle of length $\ell$ and $C_{\ell}\cup C_t$ denote the union of disjoint cycles $C_{\ell}$ and $C_t$. The planar Turán number $ex_{\mathcal{P}}(n,H)$ is known if $H=C_{\ell}\cup C_k$, where $\ell,k\in \{3,4\}$. In this paper, we determine the value $ex_{\mathcal{P}}(n,C_3\cup C_5)=\lfloor\frac{8n-13}{3}\rfloor$ and characterize the extremal graphs when $n$ is sufficiently large. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16351 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar Turán number of disjoint union of $C_3$ and $C_5$ Li, Luyi Li, Ping Yan, Guiying Zhou, Qiang Combinatorics The planar Turán number of $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Turán number of $k\geq 3$ vertex-disjoint union of cycles is the trivial value $3n-6$. Let $C_{\ell}$ denote the cycle of length $\ell$ and $C_{\ell}\cup C_t$ denote the union of disjoint cycles $C_{\ell}$ and $C_t$. The planar Turán number $ex_{\mathcal{P}}(n,H)$ is known if $H=C_{\ell}\cup C_k$, where $\ell,k\in \{3,4\}$. In this paper, we determine the value $ex_{\mathcal{P}}(n,C_3\cup C_5)=\lfloor\frac{8n-13}{3}\rfloor$ and characterize the extremal graphs when $n$ is sufficiently large. |
| title | Planar Turán number of disjoint union of $C_3$ and $C_5$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.16351 |