Bad drawings of small complete graphs

TL;DR

This paper investigates the minimal crossing configurations of small complete graphs like K_5, K_{3,3}, and K_6, providing constructions and cohomological obstructions to understand their crossing patterns.

Contribution

It introduces new constructions for drawings of K_5 and K_{3,3} with specified odd numbers of independent crossings and uses cohomology to characterize possible crossing patterns.

Findings

For K_5, drawings with odd independent crossings from 1 to 15 exist.

For K_{3,3}, similar drawings exist with odd independent crossings from 1 to 17.

For K_6, the range of possible independent crossings is from 3 to 40.

Abstract

We show that for $K_5$ (resp.~ $K_{3,3}$) there is a drawing with $i$ independent crossings, and no pair of independent edges cross more than once, provided $i$ is odd with $1\le i\le 15$ (resp.~ $1\le i\le 17$). Conversely, using the deleted product cohomology, we show that for $K_5$ and $K_{3,3}$, if $A$ is any set of pairs of independent edges, and $A$ has odd cardinality, then there is a drawing in the plane for which each element in $A$ cross an odd number of times, while each pair of independent edges not in $A$ cross an even number of times. For $K_6$ we show that there is a drawing with $i$ independent crossings, and no pair of independent edges cross more than once, if and only if $3\le i\le 40$.