Dense circuit graphs and the planar Tur\'an number of a cycle

TL;DR

This paper investigates the maximum edges in planar graphs avoiding cycles of length k, establishing bounds that are tight for large k and confirming a conjecture related to planar Turán numbers.

Contribution

It introduces the concept of circuit graphs to analyze dense planar graphs and derives tight bounds for the planar Turán number of cycles, confirming a conjecture for large cycle lengths.

Findings

Established upper bounds for x_{\u2119}(n,C_k) with a new constant D.

Proved the bounds are tight for k rom 11 upwards.

Confirmed a conjecture by Cranston et al. for large k.

Abstract

The $\textit{planar Tur\'an number}$ $\textrm{ex}_{\mathcal P}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_k$ denote the cycle of length $k$. The planar Tur\'an number $\textrm{ex}_{\mathcal P}(n,C_k)$ is known for $k\le 7$. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Tur\'an numbers. In particular, we prove that there is a constant $D$ so that $\textrm{ex}_{\mathcal P}(n,C_k) \le 3n - 6 - Dn/k^{\log_2^3}$ for all $k, n\ge 4$. When $k \ge 11$ this bound is tight up to the constant $D$ and proves a conjecture of Cranston, Lidick\'y, Liu, and Shantanam.