A Bipartite Strengthening of the Crossing Lemma

Let G = (V, E) be a graph with n vertices and m ≥ 4n edges drawn in the plane. The celebrated Crossing Lemma states that G has at least Ω(m3 /n2 ) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m2 /n2 ) other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most O(1) points. An -grid in the drawing of G is a pair E1, E2 ⊂ E of disjoint edge subsets each of size such that every edge in E1 intersects every edge in E2. If every pair of edges of G intersect in at most k points, then G contains an -grid with ≥ ckm2 /n2 , where ck > 0 only depends on k. Without any assumption on the number of points in which edges cross, we prove that G contains an -grid with = m2 /n2 polylog(m/n). If G is dense, that is, m = Θ(n2 ), our proof demonstrates that G contains an -grid with = Ω(n2 / log n). We show that this bound is best possible up to a constant factor by constructing a drawing of the complete bipartite graph Kn,...

• Type: Conference
• Year: 2007
• Where: GD
• Authors: Jacob Fox, János Pach, Csaba D. Tóth
• Keywords: Celebrated Crossing Lemma, Computer Graphics, Crossing Lemma, Edge, GD 2007