On ≤k-Edges, Crossings, and Halving Lines of Geometric Drawings of K n

Abstract

Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by $\operatorname {cr}(P)$, is the rectilinear crossing number of P. A halving line of P is a line passing through two points of P that divides the rest of the points of P in (almost) half. The number of halving lines of P is denoted by h(P). Similarly, a k-edge, 0≤k≤n/2−1, is a line passing through two points of P and leaving exactly k points of P on one side. The number of ≤k-edges of P is denoted by E ≤k (P). Let $\overline {\mathrm {cr}}(n)$, h(n), and E ≤k (n) denote the minimum of $\operatorname {cr}(P)$, the maximum of h(P), and the minimum of E ≤k (P), respectively, over all sets P of n points in general position in the plane. We show that the previously best known lower bound on E ≤k (n) is tight for k<⌈(4n−2)/9⌉ and improve it for all k≥⌈(4n−2)/9⌉. This in turn improves the lower bound on $\overline {\mathrm {cr}}(n)$ from $0.37968\binom{n}{4}+\varTheta (n^{3})$ to $\frac{277}{729}\binom{n}{4}+\varTheta (n^{3})\geq 0.37997\binom{n}{4}+\varTheta (n^{3})$. We also give the exact values of $\overline {\mathrm {cr}}(n)$ and h(n) for all n≤27. Exact values were known only for n≤18 and odd n≤21 for the crossing number, and for n≤14 and odd n≤21 for halving lines.