Planar Tur\'an number of the 7-cycle

TL;DR

This paper determines an upper bound for the maximum number of edges in large planar graphs that do not contain a 7-cycle, advancing understanding of planar Turán numbers for cycles of length 7.

Contribution

It establishes a new upper bound for the planar Turán number of the 7-cycle and proves the bound is tight for infinitely many values of n.

Findings

Upper bound of (18n/7) - 48/7 for large n

Existence of infinitely many n where the bound is tight

Progress in understanding cycle restrictions in planar graphs

Abstract

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ 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 Tur\'an 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 $n$.