The planar Tur\'an number of the seven-cycle

TL;DR

This paper establishes a sharp upper bound for the planar Turán number of the seven-cycle, confirming a conjecture and extending previous results on cycles in planar graphs.

Contribution

It provides the first sharp upper bound for $ex_ ext{P}(n,C_7)$, confirming a conjecture and also determining the bound for graphs avoiding both $K_4$ and $C_7$.

Findings

Sharp upper bound for $ex_ ext{P}(n,C_7)$: $(18/7)n - 48/7$

Bound is sharp for graphs avoiding $K_4$ and $C_7$

Confirms conjecture by Ghosh et al. on $ex_ ext{P}(n,C_k)$ for $k \\geq 7$

Abstract

The planar Tur\'an number, $ex_\mathcal{P}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_4)$ and $ex_\mathcal{P}(n,C_5)$. Later on, D. Ghosh et al. obtained sharp upper bound of $ex_\mathcal{P}(n,C_6)$ and proposed a conjecture on $ex_\mathcal{P}(n,C_k)$ for $k\geq 7$. In this paper, we give a sharp upper bound $ex_\mathcal{P}(n,C_7)\leq {18\over 7}n-{48\over 7}$, which satisfies the conjecture of D. Ghosh et al. It turns out that this upper bound is also sharp for $ex_\mathcal{P}(n,\{K_4,C_7\})$, the maximum number of edges in an $n$-vertex planar graph which does not contain $K_4$ or $C_7$ as a subgraph.