New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of Kn

• Authors: Oswin Aichholzer;Jesus Garcia;David Orden;Pedro Ramos
• Venue: Discrete & Computational Geometry
• Year: 2007

We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for $0 \leq k \leq \lfloor({n-2})/{2}\rfloor$ the number of (≤ k)-edges is at least $E_k(S) \geq 3\binom{k+2}{2} + \sum_{j=\lfloor{n}/{3}\rfloor}^k(3j-n+3),$ which, for $\lfloor {n}/{3}\rfloor\leq k\leq 0.4864n$, improves the previous best lower bound in [12]. As a main consequence, we obtain a new lower bound on the rectilinear crossing number of the complete graph or, in other words, on the minimum number of convex quadrilaterals determined by n points in the plane in general position. We show that the crossing number is at least $\left(\frac{41}{108}+\varepsilon \right) \binom{n}{4} + O(n^3) \geq 0.379688 \binom{n}{4} + O(n^3),$ which improves the previous bound of $0.37533 \binom{n}{4} + O(n^3)$ in [12] and approaches the best known upper bound $0.380559 \binom{n}{4} + \Theta(n^3)$ in [4]. The proof is based on a result about the structure of sets attaining the rectilinear crossing number, for which we show that the convex hull is always a triangle. Further implications include improved results for small values of n. We extend the range of known values for the rectilinear crossing number, namely by $\mathop{\overline{\rm cr}}\nolimits(K_{19})=1318$ and $\mathop{\overline{\rm cr}}\nolimits(K_{21})=2055$. Moreover, we provide improved upper bounds on the maximum number of halving edges a point set can have.