@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. 
	}
}