@article{FHMS20essentially4connectedplus,
title={On the circumference of essentially 4-connected planar graphs},
author={Fabrici, Igor and Harant, Jochen and Mohr, Samuel and Schmidt, Jens M},
year={2020},
journal = {Journal of Graph Algorithms and Applications},
archivePrefix = {arXiv},
eprint = {1806.09413},
volume={24},
number={1},
pages={21--46},
doi = {10.7155/jgaa.00516},
biburl = {http://samuel-mohr.de/files/bib/5.bib},
abstract = {
A planar graph is {\emph{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 {$\frac{3}{5}(n+2)$}.
}
}