@article{FHMMSZ20cycles1planar,
title = {Long cycles and spanning subgraphs of locally maximal 1-planar graphs},
journal = {Journal of Graph Theory},
author = {Fabrici, Igor and Harant, Jochen and Madaras, Tomáš and Mohr, Samuel and Soták, Roman and Zamfirescu, Carol T},
volume={95},
number={1},
pages={125--137},
year={2020},
publisher={Wiley Online Library},
doi = {10.1002/jgt.22542},
archivePrefix = {arXiv},
eprint = {1912.08028},
biburl = {http://samuel-mohr.de/files/bib/7.bib},
abstract = {Chen and Yu verified a conjecture of Moon and Moser
that there is a positive constant {$c$} such that the length of a longest cycle of each 3-connected planar graph {$G$} is at least {$c\cdot|V(G)|^{\log_32}$}. \\
A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge and it is maximal 1-planar if it is 1-planar and no edge between non-adjacent vertices can be added to keep its 1-planarity.
We will confirm the results of Moon and Moser and Chen and Yu for the class of maximal 1-planar
that
a longest cycle of 3-connected maximal 1-planar is at least {$c\cdot|V(G)|^{\log_32}$} and this bound is asymptotically optimal.
}
}