Planar Turán number of the 6-cycle

Abstract

Let ${\mathrm{ex}}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\mathrm{ex}}_{\mathcal{P}}(n,T,H)$ is the well studied function, the planar Turán number of $H$, denoted by ${\mathrm{ex}}_{\mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${\mathrm{ex}}_{\mathcal{P}}(n,C_4)$ and ${\mathrm{ex}}_{\mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${\mathrm{ex}}_{\mathcal{P}}(n,C_6)\le \frac{18(n-2)}{7}$. In this paper, we give a sharp upper bound ${\mathrm{ex}}_{\mathcal{P}}(n,C_6)\le \frac{5}{2}n-7$, for all $n\ge 18$, which improves Lan's result. We also pose a conjecture on ${\mathrm{ex}}_{\mathcal{P}}(n,C_k)$, for $k\ge 7$.

27 pages, 17 figures