Bipartite Intrinsically Knotted Graphs with 22 Edges

Hyoungjun Kim, Thomas Mattman, Seungsang Oh

    Research output: Contribution to journalArticlepeer-review

    6 Citations (Scopus)

    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.

    Original languageEnglish
    Pages (from-to)568-584
    Number of pages17
    JournalJournal of Graph Theory
    Volume85
    Issue number2
    DOIs
    Publication statusPublished - 2017 Jun

    Bibliographical note

    Publisher Copyright:
    © 2016 Wiley Periodicals, Inc.

    Keywords

    • 05C10
    • 2010 Mathematics Subject Classification: 57M25
    • 57M27

    ASJC Scopus subject areas

    • Geometry and Topology
    • Discrete Mathematics and Combinatorics

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