 
Summary: 3symmetric and 3decomposable geometric drawings of Kn
B.M. ŽAbrego
M. Cetina
S. FernŽandezMerchant
J. Lea~nos
G. Salazar§
April 30, 2008
Abstract
Even the most superficial glance at the vast majority of crossingminimal geometric drawings
of Kn reveals two hardtomiss features. First, all such drawings appear to be 3fold symmetric
(or simply 3symmetric) . And second, they all are 3decomposable, 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 3decomposable, and that there are 3symmetric
optimal constructions for all n multiple of 3. In this paper, we show that any 3decomposable
geometric drawing of Kn has at least 0.380029 n
4 + (n3
) crossings. On the other hand, we
produce 3symmetric and 3decomposable drawings that improve the general upper bound for
