Intrinsic knotting and linking of almost complete partite graphs

TL;DR

This paper classifies almost complete partite graphs based on their intrinsic linking and knotting properties, and verifies a conjecture relating intrinsic knotting and linking upon vertex removal.

Contribution

It provides a comprehensive classification of graphs close to complete partite graphs regarding intrinsic linking and knotting, and confirms a conjecture about the relationship between these properties.

Findings

Classified graphs with 0, 1, or 2 edges short of complete partite graphs for intrinsic linking and knotting.

Classified intrinsic knotting for graphs on 8 vertices.

Verified a conjecture that removing a vertex from an intrinsically knotted graph yields an intrinsically linked graph.

Abstract

We classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. For graphs in these families, we verify a conjecture presented in Adams' "The Knot Book": If a vertex is removed from an intrinsically knotted graph, one obtains an intrinsically linked graph.