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3-symmetric and 3-decomposable geometric drawings of Kn B.M. Abrego
 

Summary: 3-symmetric and 3-decomposable geometric drawings of Kn
B.M. ŽAbrego
M. Cetina
S. FernŽandez-Merchant
J. Lea~nos
G. Salazar§
April 30, 2008
Abstract
Even the most superficial glance at the vast majority of crossing-minimal geometric drawings
of Kn reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric
(or simply 3-symmetric) . And second, they all are 3-decomposable, that is, there is a triangle
T enclosing the drawing, and a balanced partition A, B, C of the underlying set of points P,
such that the orthogonal projections of P onto the sides of T show A between B and C on one
side, B between A and C on another side, and C between A and B on the third side. In fact,
we conjecture that all optimal drawings are 3-decomposable, and that there are 3-symmetric
optimal constructions for all n multiple of 3. In this paper, we show that any 3-decomposable
geometric drawing of Kn has at least 0.380029 n
4 + (n3
) crossings. On the other hand, we
produce 3-symmetric and 3-decomposable drawings that improve the general upper bound for

  

Source: Abrego, Bernardo - Department of Mathematics, California State University, Northridge
Fernandez, Silvia - Department of Mathematics, California State University, Northridge

 

Collections: Mathematics