Extremal C4-free/C5-free planar graphs

TL;DR

This paper investigates the maximum number of edges in planar graphs that avoid small cycles, specifically C4 and C5, providing tight bounds for these extremal values.

Contribution

It establishes tight upper bounds for the maximum edges in planar graphs without C4 or C5 cycles, advancing extremal graph theory in planar settings.

Findings

Derived upper bounds for ex_P(n,C4) and ex_P(n,C5).

Proved the bounds are tight for all sufficiently large n.

Abstract

We study the topic of "extremal" planar graphs, defining $\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible in a planar graph on $n$ vertices that does not contain a given graph $H$ as a subgraph. In particular,we examine the case when $H$ is a small cycle,obtaining $\mathrm{ex_{_{\mathcal{P}}}}(n,C_{4}) \leq \frac{15}{7}(n-2)$ for all $n \geq 4$ and $\mathrm{ex_{_{\mathcal{P}}}}(n,C_{5}) \leq \frac{12n-33}{5}$ for all $n \geq 11$, and showing that both of these bounds are tight.