Journal of Integer Sequences, Vol. 13 (2010), Article 10.8.5

Counting the Regions in a Regular Drawing of Kn,n

Martin Griffiths
School of Education
University of Manchester
M13 9PL
United Kingdom


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.

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(Concerned with sequences A006561 A007678 A115004 A159065.)

Received December 23 2009; revised version received March 31 2010; August 5 2010. Published in Journal of Integer Sequences, August 13 2010.

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