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Bipartite Intrinsically Knotted Graphs with 22 Edges
Author(s) -
Kim Hyoungjun,
Mattman Thomas,
Oh Seungsang
Publication year - 2017
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22091
Subject(s) - combinatorics , bipartite graph , mathematics , cograph , discrete mathematics , 1 planar graph , embedding , robertson–seymour theorem , line graph , graph , computer science , artificial intelligence
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 andE 9 + e families. Among these, the only bipartite example with no Heawood subgraph is Cousin 110 of theE 9 + 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.