Journal of Mathematical Chemistry

, Volume 50, Issue 8, pp 2272–2280 | Cite as

Face-spiral codes in cubic polyhedral graphs with face sizes no larger than 6

  • Patrick W. Fowler
  • Mohammadreza Jooyandeh
  • Gunnar Brinkmann
Original Paper


According to the face-spiral conjecture, first made in connection with enumeration of fullerenes, a cubic polyhedron can be reconstructed from a face sequence starting from the first face and adding faces sequentially in spiral fashion. This conjecture is known to be false, both for general cubic polyhedra and within the specific class of fullerenes. Here we report counterexamples to the spiral conjecture within the 19 classes of cubic polyhedra with positive curvature, i.e., with no face size larger than six. The classes are defined by triples {p 3, p 4, p 5} where p 3, p 4 and p 5 are the respective numbers of triangular, tetragonal and pentagonal faces. In this notation, fullerenes are the class {0, 0, 12}. For 11 classes, the reported examples have minimum vertex number, but for the remaining 8 classes the examples are not guaranteed to be minimal. For cubic graphs that also allow faces of size larger than 6, counterexamples are common and occur early; we conjecture that every infinite class of cubic polyhedra described by allowed and forbidden face sizes contains non-spiral elements.


Polyhedra Graphs Graph algorithms Face-spirals Face-spiral conjecture Chemical nomenclature 

Mathematics Subject Classification

05C30 05C85 52B99 92E10 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Patrick W. Fowler
    • 1
  • Mohammadreza Jooyandeh
    • 2
  • Gunnar Brinkmann
    • 3
  1. 1.Department of ChemistryUniversity of SheffieldSheffieldUK
  2. 2.Research School of Computer ScienceAustralian National UniversityCanberraAustralia
  3. 3.Applied Mathematics and Computer ScienceGhent UniversityGhentBelgium

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