Geodesic
On plane drawings of $2$-planar graphs
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XI, Tome 488 (2019), pp. 49-65 
D. V. Karpov. On plane drawings of $2$-planar graphs. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XI, Tome 488 (2019), pp. 49-65. http://geodesic.mathdoc.fr/item/ZNSL_2019_488_a2/
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     author = {D. V. Karpov},
     title = {On plane drawings of $2$-planar graphs},
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     year = {2019},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_488_a2/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is proved that any $(2k+1)$-edge connected $k$-planar graph has a plane drawing such that any two crossing edges in this drawing cross each other exactly once. It is proved that any $2$-planar graph has a plane drawing such that any two crossing edges in this drawing has no common end and cross each other exactly once. It is also proved that any $2$-planar graph has a supergraph on the same vertex set which can be drawn such that, for any vertex $v$, among every three successive edges incident to $v$, there is at least one simple edge. (An edge is called simple if it does not intersect any other edge in this drawing).

[1] K. Wagner, “Bemerkungen zum Vierfarbenproblem”, Jahresbericht der Deutschen Mathematiker-Vereinigung, 46 (1936), 26–32

[2] I. Fáry, “On straight-line representation of planar graphs”, Acta Sci. Math. (Szeged), 11 (1948), 229–233 | MR | Zbl

[3] G. Ringel, “Ein Sechsfarbenproblem auf der Kugel”, Abhandlungen aus dem Mathematischen Seminar der Universităt Hamburg, 29 (1965), 107–117 | DOI | MR | Zbl

[4] J. Pach, G. Tóth, “Graphs drawn with few crossing per edge”, Combinatorica, 17:3 (1997), 427–439 | DOI | MR | Zbl

[5] O. V. Borodin, “Reshenie zadachi Ringelya o vershinno-granevoi raskraske ploskikh grafov i o raskraske 1-planarnykh grafov”, Metody diskretnogo analiza v izuchenii realizatsii logicheskikh funktsii, 41, Institut matematiki SO AN SSSR, Novosibirsk, 1984, 12–26