Bad drawings of small complete graphs
Grant Cairns, Emily Groves, Yuri Nikolayevsky

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 (resp.~ ) there is a drawing with independent crossings, and no pair of independent edges cross more than once, provided is odd with (resp.~ ). Conversely, using the deleted product cohomology, we show that for and , if is any set of pairs of independent edges, and has odd cardinality, then there is a drawing in the plane for which each element in cross an odd number of times, while each pair of independent edges not in cross an even number of times. For we show that there is a drawing with independent crossings, and no pair of independent edges cross more than once, if and only if .
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Taxonomy
TopicsTopological and Geometric Data Analysis · Advanced Combinatorial Mathematics · Computational Geometry and Mesh Generation
