A generalized ring spiral algorithm for coding Fullerenes and other cubic polyhedra
 
P. W. Fowler^a, A. Graovac^b, T. Pisanski^c and J. Zerovnik^c,d graovac@rudjer.irb.hr ^a University of Exeter, Exeter, UK, ^b The R. Boskovic Institute, Zagreb, Croatia, ^c University of Ljubljana, Ljubljana, Slovenia, ^d University of Maribor, Maribor, Slovenia

 
The so-called ring spiral algorithm is a convenient mean for generating and representing certain fullerenes and some other cubic polyhedra. In 1993 Manolopoulos and Fowler presented a fullerene on 380 vertices without a spiral. So smaller unspirable fullerene is known. In the spring of 1997 by 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 canonical representation for arbitrary planar cubic polyhedra. Some other questions are addressed here like possible generalizations of this method to polyhedra of higher genus and to polyhedra with vertices of arbitrary valence.