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arXiv · 2507.16351

Planar Tur\'an number of disjoint union of $C_3$ and $C_5$

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

The planar Tur\'an 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 Tur\'an 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 Tur\'an 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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Luyi Li, Ping Li, Guiying Yan, Qiang Zhou. 2025-07-22. Planar Tur\'an number of disjoint union of $C_3$ and $C_5$. https://arxiv.org/abs/2507.16351

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