A new lower bound for the bipartite crossing number with applications

Abstract

Let G be a connected bipartite graph. We give a short proof, using a variation of Menger's Theorem, for a new lower bound which relates the bipartite crossing number of G, denoted by bcr(G), to the edge connectivity properties of G. The general lower bound implies a weaker version of a very recent result, establishing a bisection-based lower bound on bcr(G) which has algorithmic consequences. Moreover, we show further applications of our general method to estimate bcr(G) for “well structured” families of graphs, for which tight isoperimetric inequalities are available. For hypercubes and two-dimensional meshes, the upper bounds (asymptotically) are within multiplicative factors of 4 and 2, from the lower bounds, respectively. The general lower bound also implies a lower bound involving eigenvalues of G.