Bipartite Intrinsically Knotted Graphs with 22 Edges |
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| Authors: | Hyoungjun Kim Thomas Mattman Seungsang Oh |
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| Institution: | 1. INSTITUTE OF MATHEMATICAL SCIENCES, EWHA WOMANS UNIVERSITY, SEOUL, KOREA;2. DEPARTMENT OF MATHEMATICS AND STATISTICS, CALIFORNIA STATE UNIVERSITY, CHICO, CA;3. DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY, SEOUL, KOREAContract grant sponsor: National Research Foundation of Korea (NRF);4. contract grant sponsor: Korea government (MSIP);5. contract grant number: NRF‐2014R1A2A1A11050999 (to S.O.). |
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| 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  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  families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the  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. |
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| Keywords: | 2010 Mathematics Subject Classification: 57M25 57M27 05C10 |
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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 
families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of the 
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.