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 $\operatorname{cr}_D(u,v)$ 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 $\operatorname{cr}_D(u,v)<Z(p)$ 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 $\mathcal D(p,\Sigma)$.