A Bipartite Strengthening of the Crossing Lemma

BASIC DEFINITIONS: crossing pair of curves and common interior point between two arcs. crossing number cr(G) minimum number of crossings in a drawing of G l-grid is a pair two disjoint edge subset E1, E2 ⊂ E of a drawing of a graph G = (V, E), such that |E1| = |E2| = l and every edge in E1 crosses every edge in E2 bi-clique complete bipartite graph where vertex classes differ in size by at most 1 x-monotone curve a curve that intersect every vertical line in at most one point x-monotone drawing drawing of graph G such that every edge is mapped to an x-monotone curve bisection width b(G) the smallest nonnegative integer such that there is a partition of the vertex set V = V1 ∪∗ V2 with 13 |V | ≤ Vi ≤ 2 3 |V | for i = 1, 2, and |E(V1, V2)| = b(G).