Geodesic
Counting the regions in a regular drawing of \(K_{n,n}\)
Journal of integer sequences, Tome 13 (2010) no. 8
We calculate here both exact and asymptotic formulas for the number of regions enclosed by the edges of a regular drawing of the complete bipartite graph $K_{n,n}$. This extends the work of Legendre, who considered the number of crossings arising from such a graph. We also show that the ratio of regions to crossings tends to a rational constant as $n$ increases without limit.
Classification : 05C62, 05A15, 05A16, 11A05
Keywords: complete bipartite graph, crossing, greatest common divisor, region 
@article{JIS_2010__13_8_a1,
     author = {Griffiths,  Martin},
     title = {Counting the regions in a regular drawing of {\(K_{n,n}\)}},
     journal = {Journal of integer sequences},
     year = {2010},
     volume = {13},
     number = {8},
     zbl = {1247.05110},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_8_a1/}
}
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Griffiths,  Martin. Counting the regions in a regular drawing of \(K_{n,n}\). Journal of integer sequences, Tome 13 (2010) no. 8. http://geodesic.mathdoc.fr/item/JIS_2010__13_8_a1/