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Four triangle-free intrinsically knotted graphs with 22 edges

  • Hyoungjun Kim
  • Thomas W. Mattman
  • Seungsang Oh
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Abstract

An intrinsically knotted graph is one for which every spatial embedding contains a nontrivially knotted cycle. Classifying such graphs is a central problem in spatial graph theory. It is known that every intrinsically knotted graph has at least 21 edges, and the case of 21 edges has been completely resolved. For 22 edges, however, the classification remains incomplete. In particular, exactly eight triangle-free examples with a vertex of degree at least 5 are known, leaving only the case in which all vertices have degree 3 or 4. In this paper, we introduce a method for detecting intrinsic knottedness based on induced subgraphs obtained by deleting pairs of vertices. Using this method, we classify all triangle-free intrinsically knotted graphs with 22 edges having eight vertices of degree~4 and four of degree~3. We prove that there are exactly four: Cousins 43, 105, and 109 in the E9 + e family and the graph H12 + e in the H9 + e family.

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Cite

BibTeXRIS
BibTeX
@misc{kim2026four,
  title = {{Four triangle-free intrinsically knotted graphs with 22 edges}},
  author = {Hyoungjun Kim and Thomas W. Mattman and Seungsang Oh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08891},
  eprint = {2609.08891},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08891},
}
RIS
TY  - UNPB
TI  - Four triangle-free intrinsically knotted graphs with 22 edges
AU  - Hyoungjun Kim
AU  - Thomas W. Mattman
AU  - Seungsang Oh
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08891
UR  - https://arxiv.org/abs/2609.08891
AB  - An intrinsically knotted graph is one for which every spatial embedding contains a nontrivially knotted cycle. Classifying such graphs is a central problem in spatial graph theory. It is known that every intrinsically knotted graph has at least 21 edges, and the case of 21 edges has been completely resolved. For 22 edges, however, the classification remains incomplete. In particular, exactly eight triangle-free examples with a vertex of degree at least 5 are known, leaving only the case in which all vertices have degree 3 or 4. In this paper, we introduce a method for detecting intrinsic knottedness based on induced subgraphs obtained by deleting pairs of vertices. Using this method, we classify all triangle-free intrinsically knotted graphs with 22 edges having eight vertices of degree~4 and four of degree~3. We prove that there are exactly four: Cousins 43, 105, and 109 in the E_9 + e family and the graph H_12 + e in the H_9 + e family.
ER  -