TY - JOUR
T1 - Bipartite Intrinsically Knotted Graphs with 22 Edges
AU - Kim, Hyoungjun
AU - Mattman, Thomas
AU - Oh, Seungsang
N1 - Funding Information:
The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education(2009-0093827).
Publisher Copyright:
© 2016 Wiley Periodicals, Inc.
PY - 2017/6
Y1 - 2017/6
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.
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.
KW - 05C10
KW - 2010 Mathematics Subject Classification: 57M25
KW - 57M27
UR - http://www.scopus.com/inward/record.url?scp=84990934106&partnerID=8YFLogxK
U2 - 10.1002/jgt.22091
DO - 10.1002/jgt.22091
M3 - Article
AN - SCOPUS:84990934106
SN - 0364-9024
VL - 85
SP - 568
EP - 584
JO - Journal of Graph Theory
JF - Journal of Graph Theory
IS - 2
ER -