A Generalized Ring Spiral Algorithm for CodingFullerenes and other Cubic Polyhedra 1

The so-called ring spiral algorithm is a convenient means for generating and representing certain fullerenes and some other cubic poly-hedra. In 1993 Manolopoulos and Fowler presented a fullerene on 380 vertices without a spiral. No smaller unspirable fullerene is known. In the spring of 1997, using computer, Gunnar Brinkmann found the smallest cubic polyhedron without a spiral. It has only 18 vertices. Here we generalize the ring spiral approach in order to obtain a canon-ical representation for arbitrary planar cubic polyhedra. Some other questions are addressed: for instance possible generalization of this method to polyhedra of higher genus and to polyhedra with vertices of arbitrary valency.