Circumference of essentially 4-connected planar triangulations

TL;DR

This paper proves that essentially 4-connected maximal planar graphs on n vertices always contain a cycle of length at least two-thirds of n plus a small constant, and this bound is proven to be tight.

Contribution

It establishes a sharp lower bound on the circumference of essentially 4-connected maximal planar graphs, advancing understanding of their cycle structure.

Findings

Every such graph has a cycle of length at least 2/3 of n plus 4.

The proven bound is tight, meaning it cannot be improved.

The result extends known cycle length bounds in planar graphs.

Abstract

A $3$-connected graph $G$ is essentially $4$-connected if, for any $3$-cut $S\subseteq V(G)$ of $G$, at most one component of $G-S$ contains at least two vertices. We prove that every essentially $4$-connected maximal planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2}{3}(n+4)$; moreover, this bound is sharp.