Bipartite intrinsically knotted graphs with 23 edges

TL;DR

This paper classifies bipartite intrinsically knotted graphs with 23 edges, showing that no minor minimal bipartite intrinsically knotted graph exists at this edge count.

Contribution

It identifies all bipartite intrinsically knotted graphs with 23 edges and demonstrates the absence of minor minimal bipartite intrinsically knotted graphs at this size.

Findings

Six bipartite intrinsically knotted graphs with 23 edges identified.

Four contain the Heawood graph, two contain Cousin 110.

No minor minimal bipartite intrinsically knotted graph exists with 23 edges.

Abstract

A graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that there are exactly 14 intrinsically knotted graphs with 21 edges, in which the Heawood graph is the only bipartite graph. The authors showed that there are exactly two graphs with at most 22 edges that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110 of the $E_9+e$ family. In this paper we show that there are exactly six bipartite intrinsically knotted graphs with 23 edges so that every vertex has degree 3 or more. Four among them contain the Heawood graph and the other two contain Cousin 110 of the $E_9+e$ family. Consequently, there is no minor minimal intrinsically knotted graph with 23 edges that is bipartite.