Bipartite crossing numbers of meshes and hypercubes

Abstract

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 edge were placed. The bipartite crossing number of G, denoted by bcr(G) is the minimum number of crossings of edges over all bipartite drawings of G. We develop a new lower bound method for estimating bcr(G). It relates bipartite crossing numbers to edge isoperimetric inequalities and Laplaclan eigenvalues of graphs. We apply the method, which is suitable for "well structured" graphs, to hypercubes and 2-dimensional meshes. E.g. for the n-dimensional hypercube graph we get n4n-2-O(4n) ≤ bcr(Qn) < n4n-1. We also consider a more general setting of the method which uses eigenvalues, but as a trade-off for generality, often gives weaker results.