A new construction for planar Tur\'an number of cycle

TL;DR

This paper establishes a new lower bound for the planar Turán number of cycles, demonstrating that a previously conjectured upper bound is essentially tight, thereby advancing understanding of extremal planar graphs without certain cycles.

Contribution

The authors provide a new construction that confirms the conjectured upper bound for the planar Turán number of cycles is nearly optimal, resolving an open question in extremal graph theory.

Findings

New lower bound matches the conjectured upper bound asymptotically.

Confirms the conjecture by Cranston et al. is essentially best possible.

Advances understanding of cycle-free planar graphs with maximum edges.

Abstract

The planar Tur\'an number ${\rm ex}_{\mathcal{P}}(n,C_k)$ 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, Lidick\'y, Liu and Shantanam \cite{2021Planar}, and independently Lan and Song \cite{LanSong} showed that ${\rm ex}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{Cn}{k}$ for large $n$. Moreover, Cranston et al. conjectured that ${\rm ex}_{\mathcal{P}}(n,C_k)\le 3n-6-\frac{Dn}{k^{lg_23}}$ when $n$ is large. In this note, we prove that ${\rm ex}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{6\cdot 3^{lg_23}n}{k^{lg_23}}$ for every $k$. It implies Cranston et al.'s conjecture is essentially best possible.