A New Construction for the Planar Turán Number of Cycles

The planar Turán number \(\textrm{ex}_{\mathcal {P}}(n,C_k)\) ${\text{ex}}_{\mathcal{P}}(n,C_k)$ is the maximum number of edges in an n-vertex planar graph not containing a cycle of length k. Let \(k\ge 11\) $k\ge 11$ and c, d be constants. Cranston et al., and independently Lan and Song showed that \(\textrm{ex}_{\mathcal {P}}(n,C_k)\ge 3n-6- cn/k\) ${\text{ex}}_{\mathcal{P}}(n,C_k)\ge 3n-6-cn/k$ holds for large n. Moreover, Cranston et al. conjectured that \(\textrm{ex}_{\mathcal {P}}(n,C_k)\le 3n-6- dn/k^{\log _2 3}\) ${\text{ex}}_{\mathcal{P}}(n,C_k)\le 3n-6-dn/k^{{log}_{2}3}$ when n is large. In this note, we prove that \(\textrm{ex}_{\mathcal {P}}(n,C_k)\ge 3n-6-6\cdot 3^{\log _23}n/k^{\log _2 3}\) ${\text{ex}}_{\mathcal{P}}(n,C_k)\ge 3n-6-6·3^{{log}_{2}3}n/k^{{log}_{2}3}$ holds for every \(k\ge 7\) $k\ge 7$ . This implies that the conjecture of Cranston et al. is essentially best possible.