Jesús Leaños

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

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

A central approach to bound the number of crossings in a generalized configuration

\operatorname {cr}(P)

The maximum number of halving lines and the rectilinear crossing number of Kn for n⩽27

, 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

The crossing number of a projective graph is quadratic in the face{width

\overline {\mathrm {cr}}(n)

Regularity and Planarity of Token Graphs

, h(n), and E≤k(n) denote the minimum of

Simple Euclidean Arrangements with No (≥ 5)-Gons

\operatorname {cr}(P)

Convexifying Monotone Polygons while Maintaining Internal Visibility

, 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

On the Pseudolinear Crossing Number

\overline {\mathrm {cr}}(n)

The packing number of the double vertex graph of the path graph

from

Characterizing all graphs with 2-exceptional edges

0.37968\binom{n}{4}+\varTheta (n^{3})