On the Circumference of Essentially 4-connected Planar Graphs

TL;DR

This paper improves the lower bound on the cycle length in essentially 4-connected planar graphs from 3/5 to 5/8 of the number of vertices, advancing understanding of their structural properties.

Contribution

It establishes a new, tighter lower bound on the circumference of essentially 4-connected planar graphs, enhancing previous results.

Findings

Proves every such graph has a cycle of length at least 5/8 of n+2.

Improves the previous lower bound of 3/5(n+2).

Contributes to the theory of cycle lengths in planar graphs.

Abstract

A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2n+4}{5}$, and this result has recently been improved multiple times. In this paper, we prove that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{5}{8}(n+2)$. This improves the previously best-known lower bound $\frac{3}{5}(n+2)$.