New upper bounds for the crossing numbers of crossing-critical graphs

Bounds for Convex Crossing Numbers

Crossing Numbers: Bounds and Applications

We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. 1 aim is to emphasize the more general results or those results which have an algorith-mic avor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discret...

Improved Lower Bounds on Book Crossing Numbers of Complete Graphs

A book with k pages consists of a straight line (the spine) and k half-planes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The k-page crossing number νk(G) of a graph G is the minimu...

Crossing numbers of random graphs

An order type of the points x1, x2, ..., xn in the plane (with no three colinear) is a list of orientations of all triplet xixjxk, i < j < k. Let X be the set of all order types of the points x1, ..., xn in the plane. For any graph G with vertices v1, ..., vn let lin-crξ(G) denote the number of crossings in the straight line drawing of G, where vi is placed at xi in the plane and x1, ..., xn ha...

Crossing numbers of imbalanced graphs

The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. According to the Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi [ACNS82] and Leighton [L83], the crossing number of any graph with n vertices and e > 4n edges is at least constant times e/n. Apart from the value of the constant, this bound cannot be improved. We establish some...