@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. } }