@article{FHMS20essentially4connected,
title = {Longer cycles in essentially 4-connected planar graphs},
author = {Fabrici, Igor and Harant, Jochen and Mohr, Samuel and Schmidt, Jens M},
journal={Discussiones Mathematicae Graph Theory},
volume={40},
number={1},
pages={269--277},
year={2020},
publisher={Sciendo},
doi = {10.7151/dmgt.2133},
archivePrefix = {arXiv},
eprint = {1710.05619},
biburl = {http://samuel-mohr.de/files/bib/3.bib},
abstract = {A planar 3-connected graph {$G$} is called {\emph{essentially $4$-connected}} if, for every 3-separator {$S$}, at least one of the two components of {$G-S$} is an isolated vertex. Jackson and Wormald proved that the length {$\mathrm{circ}(G)$} of a longest cycle of any essentially 4-connected planar graph {$G$} on {$n$} vertices is at least {$\frac{2n+4}{5}$} and Fabrici, Harant and {Jendro{\v{l}}} improved this result to {$\mathrm{circ}(G)\geq \frac{1}{2}(n+4)$}.
In the present paper, we prove that an essentially 4-connected planar graph on {$n$} vertices contains a cycle of length at least {$\frac{3}{5}(n+2)$} and that such a cycle can be found in time {$O(n^2)$}.}
}