Monotone crossing number of complete graphs

Abstract

In 1958, Hill conjectured that the minimum number of crossings in a drawing of Kn is exactly Z(n) = 1/4 n-1/2/2 n−2/2 n−3/2. Generalizing the result by Ábrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by x-monotone curves. In fact, our proof shows that the conjecture remains true for xmonotone drawings in which adjacent edges do not cross and we count only pairs of edges which cross odd number of times. We also discuss a combinatorial characterization of these drawings.