Drawing disconnected graphs on the Klein bottle

TL;DR

This paper proves that when drawing two disjoint graphs on the Klein bottle, they must be separated to achieve the minimal crossing number, highlighting a unique topological constraint.

Contribution

It establishes a novel crossing number property for disjoint graphs on the Klein bottle, contributing to topological graph theory.

Findings

Disjoint graphs on the Klein bottle must be drawn separately for minimal crossings.

Separation reduces the crossing number in Klein bottle drawings.

Provides new insights into graph embeddings on non-orientable surfaces.

Abstract

We prove that two disjoint graphs must always be drawn separately on the Klein bottle, in order to minimize the crossing number of the whole drawing.