Bipartite intrinsically knotted graphs with 22 edges

TL;DR

This paper classifies all bipartite intrinsically knotted graphs with up to 22 edges, identifying only two such graphs: the Heawood graph and Cousin 110, thus completing the known list.

Contribution

It provides a complete classification of minor minimal bipartite intrinsically knotted graphs with at most 22 edges, confirming only two exist.

Findings

Only two bipartite intrinsically knotted graphs with ≤22 edges: Heawood graph and Cousin 110.

Previously known examples include graphs with KS minors and certain families; this work confirms the complete list.

No other bipartite intrinsically knotted graphs with ≤22 edges exist beyond these two.

Abstract

A graph is intrinsically knotted if every embedding contains a knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that the KS graphs, $K_7$ and the 13 graphs obtained from $K_7$ by $\nabla Y$ moves, are the only minor minimal intrinsically knotted graphs with 21 edges. This set includes exactly one bipartite graph, the Heawood graph. In this paper we classify the intrinsically knotted bipartite graphs with at most 22 edges. Previously known examples of intrinsically knotted graphs of size 22 were those with KS graph minor and the 168 graphs in the $K_{3,3,1,1}$ and $E_9+e$ families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the $E_9+e$ family. We show that, in fact, this is a complete listing. That is, there are exactly two graphs of size at most 22 that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110.