The Complement Problem for Linklessly Embeddable Graphs

TL;DR

This paper classifies maximal linklessly embeddable graphs up to 11 vertices, explores properties of their complements, and verifies a conjecture related to the Colin de Verdière invariant for small graphs.

Contribution

It provides a complete classification of maximal linklessly embeddable graphs up to size 11 and verifies a related Nordhaus-Gaddum conjecture for these graphs.

Findings

All maximal linklessly embeddable graphs of order up to 11 are identified.

For every graph of order 11, either it or its complement is intrinsically linked.

An example of an 11-vertex graph with both it and its complement being $K_6$-minor free.

Abstract

We find all maximal linklessly embeddable graphs of order up to 11, and verify that for every graph $G$ of order 11 either $G$ or its complement $cG$ is intrinsically linked. We give an example of a graph $G$ of order 11 such that both $G$ and $cG$ are $K_6$-minor free. We provide minimal order examples of maximal linklessly embeddable graphs that are not triangular or not 3-connected. We prove a Nordhaus-Gaddum type conjecture on the Colin de Verdi\`ere invariant for graphs on at most 11 vertices. We give a description of the programs used in the search.