Bounding the Number of Non-duplicates of the q-Side in Simple Drawings of $$K_{p,q}$$

Abstract

The number \(Z(n):=\lfloor n/2\rfloor \lfloor (n-1)/2\rfloor\) is the smallest number of crossings in a simple planar drawing of \(K_{2,n}\) in which both vertices on the 2-side have the same clockwise rotation. For two vertices u, v on the q-side of a simple drawing of \(K_{p,q}\), let \({\text {cr}}_D(u,v)\) denote the total number of crossings that edges incident with u have with edges incident with v. We show that in any simple drawing D of \(K_{p,q}\) in a surface \(\Sigma\) the number of pairs of vertices on the q-side of \(K_{p,q}\) having \({\text {cr}}_D(u,v)<Z(p)\) is bounded as a function of p and \(\Sigma\). As a consequence, we also show that, for a fixed integer p and surface \(\Sigma\), there exists a finite set of drawings \({\mathcal {D}}(p,\Sigma )\) of complete bipartite graphs such that, for each q, a crossing-minimal drawing of \(K_{p,q}\) can be obtained by “duplicating vertices” in some drawing from \({\mathcal {D}}(p,\Sigma )\).