Geodesic
Upper bound on the number of edges of an almost planar bipartite graph
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 406 (2012), pp. 12-30

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Let $G$ be a bipartite graph without loops and multiple edges on $v\ge4$ vertices, which can be drawn on the plane such that any edge intersects at most one other edge. We prove that such graph has at most $3v-8$ edges for even $v\ne6$ and at most $3v-9$ edges for odd $v$ and $v=6$. For all $v\ge4$ examples showing that these bounds are tight are constructed. In the end of the paper, we discuss a question about drawings of complete bipartite graphs on the plane such that any edge intersects at most one other edge. 
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     author = {D. V. Karpov},
     title = {Upper bound on  the number of edges of an almost planar bipartite graph},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {12--30},
     publisher = {mathdoc},
     volume = {406},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_406_a1/}
}
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D. V. Karpov. Upper bound on  the number of edges of an almost planar bipartite graph. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 406 (2012), pp. 12-30. http://geodesic.mathdoc.fr/item/ZNSL_2012_406_a1/