@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 {$\frac{3}{5}(n+2)$}.
  }
}