A new construction for planar Turán number of cycle

Abstract

The planar Turán number ${\mathrm{ex}}_{\mathcal{P}}(n,C_k)$ is the largest number of edges in an $n$-vertex planar graph with no cycle of length $k$. Let $k\ge 11$ and $C,D$ be constants. Cranston, Lidický, Liu and Shantanam \cite{2021Planar}, and independently Lan and Song \cite{LanSong} showed that ${\mathrm{ex}}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{Cn}{k}$ for large $n$. Moreover, Cranston et al. conjectured that ${\mathrm{ex}}_{\mathcal{P}}(n,C_k)\le 3n-6-\frac{Dn}{k^{l{g}_{2}3}}$ when $n$ is large. In this note, we prove that ${\mathrm{ex}}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{6\cdot 3^{l{g}_{2}3}n}{k^{l{g}_{2}3}}$ for every $k$. It implies Cranston et al.'s conjecture is essentially best possible.

5 pages, 1 figures