We prove that the minimum number of convex quadrilaterals determined by n points in general position in the plane – or in other words, the rectilinear crossing number of the complete graph Kn – is at least ( 38 + 10 −5) ( n 4 ) +O(n). Our main tool is a lower bound on the number of (≤ k)-sets of the point set: we show that for every k ≤ n/2, there are at least 3 ( k+1 2 ) subsets of size at mos...

The crossing number (G) of a graph G is the smallest integer such that there is a drawing for G with (G) crossings of edges. Let Q n denote the n{dimensional

We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly–light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k–crossing–critical graph has crossing number at most 2k + 23.

In this paper we generalize the concept of alternating knots to alternating graphs and show that every abstract graph has a spatial embedding that is alternating. We also prove that every spatial graph is a subgraph of an alternating graph. We define n-composition for spatial graphs and generalize the results of Menasco on alternating knots to show that an alternating graph is n-composite for n...

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram E with n chords is called an n-crossing if all chords of E are mutually crossing. A chord diagram E is called nonintersecting if E contains no 2-crossing. For a chord diagram E having a 2-crossing S = {x1x3, x2x4}, the expansion of E with respect to S is to replace E with E1 = (E\S...

Let G = (V0; V1; E) be a connected bipartite graph, where V0; V1 is the bipartition of the vertex set V (G) into independent sets. A bipartite drawing of G consists of placing the vertices of V0 and V1 into distinct points on two parallel lines x0 ; x1, respectively, and then drawing each edge with one straight line segment which connects the points of x0 and x1 where the endvertices of the edg...

The crossing number cr(G) of a graphG is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [RT93] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that cr(G− e) ≥ k− c √ k. They showed only that G always has an edge e with cr(G − e) ≥ 2 5 cr(G) − O(1). We prove that for every fixed ǫ > 0, t...