Complete Minors in Complements of Non-Separating Planar Graphs

TL;DR

This paper proves that the complement of any non-separating planar graph of a certain size contains a complete minor, establishing the minimal order for this property and exploring related graph properties.

Contribution

It establishes the minimal order for non-separating planar graphs whose complements contain a complete minor and provides examples illustrating the necessity of conditions.

Findings

Complement of non-separating planar graphs of order 2n-3 contains a K_n minor.

Order 2n-3 is minimal for this property.

Complements of planar graphs of order 11 are intrinsically knotted.

Abstract

We prove that the complement of any non-separating planar graph of order $2n-3$ contains a $K_n$ minor, and argue that the order $2n-3$ is lowest possible with this property. To illustrate the necessity of the non-separating hypothesis, we give an example of a planar graph of order 11 whose complement does not contain a $K_7$ minor. We argue that the complements of planar graphs of order 11 are intrinsically knotted. We compute the Hadwiger numbers of complements of wheel graphs.