A bipartite strengthening of the Crossing Lemma

Abstract

The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m3/n 2) 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. We prove for every that every graph G with n vertices and m ≥ 3n edges drawn in the plane such that any two edges intersect in at most k points has two disjoint subsets of edges, E 1 and E 2, each of size at least , such that every edge in E 1 crosses all edges in E 2, where ck > 0 only depends on k. This bound is best possible up to the constant c k for every . We also prove that every graph G with n vertices and m ≥ 3n edges drawn in the plane with x-monotone edges has disjoint subsets of edges, E 1 and E 2, each of size Ω(m 2/ (n 2 polylog n)), such that every edge in E 1 crosses all edges in E 2. On the other hand, we construct x-monotone drawings of bipartite dense graphs where the largest such subsets E 1 and E 2 have size O(m 2/(n 2 log(m/n))). © 2008 Springer-Verlag Berlin Heidelberg.