Planar Turán number of disjoint union of $C_3$ and $C_5$ Article Swipe

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Luyi Li , Ping Li , Guiying Yan , Qiang Zhou ·

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2507.16351

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.

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2507.16351
PDF: https://arxiv.org/pdf/2507.16351
OA Status: green
OpenAlex ID: https://openalex.org/W4414415092

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DOI: https://doi.org/10.48550/arxiv.2507.16351

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Title: Planar Turán number of disjoint union of $C_3$ and $C_5$

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-07-22

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Authors: Luyi Li, Ping Li, Guiying Yan, Qiang Zhou

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Landing page: https://arxiv.org/abs/2507.16351

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