### 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 the KS graphs, K_{7} and the 13 graphs obtained from K_{7} by ∇Y moves, are the only minor minimal intrinsically knotted graphs with 21 edges [1, 9, 11, 12]. This set includes exactly one bipartite graph, the Heawood graph. In this article 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 E9+e families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the E9+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. Copyright

Original language | English |
---|---|

Journal | Journal of Graph Theory |

DOIs | |

Publication status | Accepted/In press - 2016 |

### Fingerprint

### Keywords

- 05C10
- 2010 Mathematics Subject Classification: 57M25
- 57M27

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*. https://doi.org/10.1002/jgt.22091

**Bipartite Intrinsically Knotted Graphs with 22 Edges.** / Kim, Hyoungjun; Mattman, Thomas; Oh, Seung Sang.

Research output: Contribution to journal › Article

*Journal of Graph Theory*. https://doi.org/10.1002/jgt.22091

}

TY - JOUR

T1 - Bipartite Intrinsically Knotted Graphs with 22 Edges

AU - Kim, Hyoungjun

AU - Mattman, Thomas

AU - Oh, Seung Sang

PY - 2016

Y1 - 2016

N2 - 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 the KS graphs, K7 and the 13 graphs obtained from K7 by ∇Y moves, are the only minor minimal intrinsically knotted graphs with 21 edges [1, 9, 11, 12]. This set includes exactly one bipartite graph, the Heawood graph. In this article 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 K3, 3, 1, 1 and E9+e families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the E9+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. Copyright

AB - 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 the KS graphs, K7 and the 13 graphs obtained from K7 by ∇Y moves, are the only minor minimal intrinsically knotted graphs with 21 edges [1, 9, 11, 12]. This set includes exactly one bipartite graph, the Heawood graph. In this article 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 K3, 3, 1, 1 and E9+e families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the E9+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. Copyright

KW - 05C10

KW - 2010 Mathematics Subject Classification: 57M25

KW - 57M27

UR - http://www.scopus.com/inward/record.url?scp=84990934106&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990934106&partnerID=8YFLogxK

U2 - 10.1002/jgt.22091

DO - 10.1002/jgt.22091

M3 - Article

AN - SCOPUS:84990934106

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

ER -