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

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• 作者：Patrick W. Fowler (1) P.W.Fowler@sheffield.ac.uk
  Mohammadreza Jooyandeh (2) mohammadreza@jooyandeh.info
  Gunnar Brinkmann (3) Gunnar.Brinkmann@ugent.be
• 关键词：Polyhedra – ; Graphs – ; Graph algorithms – ; Face ; spirals – ; Face ; spiral conjecture – ; Chemical nomenclature
• 刊名：Journal of Mathematical Chemistry
• 出版年：2012
• 出版时间：September 2012
• 年：2012
• 卷：50
• 期：8
• 页码：2272-2280
• 全文大小：516.6 KB
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• 作者单位：1. Department of Chemistry, University of Sheffield, Sheffield, S3 7HF UK2. Research School of Computer Science, Australian National University, Canberra, ACT 0200, Australia3. Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, Ghent, 9000 Belgium
• ISSN：1572-8897

文摘

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.