Bounding the number of non-duplicates of the $q$-side in simple drawings of $K_{p,q}$

TL;DR

This paper investigates the structure of simple drawings of complete bipartite graphs on surfaces, establishing bounds on non-duplicate pairs of vertices based on crossing numbers and showing how minimal drawings can be generated from a finite set.

Contribution

It introduces bounds on the number of non-duplicate vertex pairs in simple drawings of $K_{p,q}$ and demonstrates that crossing-minimal drawings can be derived from a finite set via vertex duplication.

Findings

Bound on pairs of vertices with low crossing number in any drawing.

Existence of a finite set of base drawings for each fixed p and surface.

Crossing-minimal drawings obtained by duplicating vertices in base drawings.

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 $\mathcal D(p,\Sigma)$.