Tangled piecewise-linear embeddings of trivalent graphs

Michael O'Keeffe, Michael M.J. Treacy

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


A method is described for generating and exploring tangled piecewise-linear embeddings of trivalent graphs under the constraints of point-group symmetry. It is shown that the possible vertex-transitive tangles are either graphs of vertextransitive polyhedra or bipartite vertex-transitive nonplanar graphs. One tangle is found for 6 vertices, three for 8 vertices (tangled cubes), seven for 10 vertices, and 21 for 12 vertices. Also described are four isogonal embeddings of pairs of cubes and 12 triplets of tangled cubes (16 and 24 vertices, respectively). Vertex 2-transitive embeddings are obtained for tangled trivalent graphs with 6 vertices (two found) and 8 vertices (45 found). Symmetrical tangles of the 10- vertex Petersen graph and the 20-vertex Desargues graph are also described. Extensions to periodic tangles are indicated. These are all interesting and viable targets for molecular synthesis.

Original languageEnglish (US)
Pages (from-to)128-138
Number of pages11
JournalActa Crystallographica Section A: Foundations and Advances
StatePublished - Mar 1 2022


  • Tangled molecular targets
  • Tangled structures
  • Trivalent graphs

ASJC Scopus subject areas

  • Structural Biology
  • Biochemistry
  • General Materials Science
  • Condensed Matter Physics
  • Physical and Theoretical Chemistry
  • Inorganic Chemistry


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