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 -