S. Fernández-Merchant †

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 ) + Θ(n) crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the general upper bound for the rectilinear crossing number of Kn to 0.380488 ( n 4 ) +Θ(n). We also give explicit 3-symmetric and 3-decomposable constructions for n < 100 that are at least as good as those previously known.