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arXiv · 2306.13594

Planar Turán number of the 7-cycle

Ruilin Shi·Zach Walsh·Xingxing Yu
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Abstract

The $\textit{planar Turán number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_{\ell}$ denote the cycle of length $\ell$. The planar Turán number $\textrm{ex}_{\mathcal P}(n,C_{\ell})$ behaves differently for $\ell\le 10$ and for $\ell\ge 11$, and it is known when $\ell \in \{3,4,5,6\}$. We prove that $\textrm{ex}_{\mathcal P}(n,C_7) \le \frac{18n}{7} - \frac{48}{7}$ for all $n > 38$, and show that equality holds for infinitely many integers $n$.

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Ruilin Shi, Zach Walsh, Xingxing Yu. 2023-06-23. Planar Turán number of the 7-cycle. https://arxiv.org/abs/2306.13594

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