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Intersection of Curves and Crossing Number of
C
m
× C
n
on Surfaces
| |
|
Intersection of Curves and Crossing Number of C
m
× C
n
on Surfaces
F. Shahrokhi1, O. Sýkora2, L. A. Székely3 and I. Vrt'o2
| (1) |
Department of Computer Science, University of North Texas, P.O.Box 13886, Denton, TX 76203-3886, USA, US |
| (2) |
Institute for Informatics, Slovak Academy of Sciences, P.O.Box 56, 840 00 Bratislava, Slovak Republic, SK |
| (3) |
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA, US |
Abstract. Let be two families of closed curves on a surface , such that , each curve in intersects each curve in , and no point of is covered three times. When is the plane, the projective plane, or the Klein bottle, we prove that the total number of intersections in is at least 10mn/9 , 12mn/11 , and mn+10
-13
m
2
, respectively. Moreover, when m is close to n , the constants are improved. For instance, the constant for the plane, 10/9 , is improved to 8/5 , for n ≤ 5(m-1)/4 . Consequently, we prove lower bounds on the crossing number of the Cartesian product of two cycles, in the plane, projective
plane, and the Klein bottle. All lower bounds are within small multiplicative factors from easily derived upper bounds. No
general lower bound has been previously known, even on the plane.
Received January 20, 1996, and in revised form October 21, 1996.
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