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, 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
| Original language | English |
|---|---|
| Journal | Journal of Graph Theory |
| DOIs | |
| Publication status | Accepted/In press - 2016 |
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Keywords
- 05C10
- 2010 Mathematics Subject Classification: 57M25
- 57M27
ASJC Scopus subject areas
- Geometry and Topology
Cite this
Bipartite Intrinsically Knotted Graphs with 22 Edges. / Kim, Hyoungjun; Mattman, Thomas; Oh, Seung Sang.
In: Journal of Graph Theory, 2016.Research output: Contribution to journal › Article
}
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 -