Face-Spiral Codes in Cubic Polyhedral Graphs with Face Sizes no Larger Than 6

Abstract

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 {p3, p4, p5} where p3, p4 and p5 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.