In this paper, we give a very efficient and completely reliable method to enumerate all combinatorial possibilities for fullerene structures. The method is based on a top-down di¨ide and conquer approach and can easily be generalized also for other kinds of spherical structures. A computer program based on this approach was used to produce complete lists of fullerene structures with up to 162 atoms. ᮊ 1998 Academic Press Presently, methods for reliable and efficient enumeration of fullerene Ž w x. isomers are a much discussed topic cf. 1᎐5 . All published procedures use a bottom-up strategy: starting from a small subconfiguration, fullerene structures are generated by stepwise enlarging this subconfiguration in all conceivable, or in some particular ways assumed to be sufficient, e.g., by using one or the other variant of the so-called spiral algorithm. Unfortunately, these methods easily meet prohibitive time constraints. So implementations often try to reduce complexity by shortcuts which then endanger reliability. Up to now, no method was known that could guarantee complete lists of fullerenes and was fast enough to be applicable for, say, 80 C-atoms.
In this note, we present a new approach to the fullerene enumeration problem which is absolutely reliableᎏthat is, it can guarantee complete listsᎏand simultaneously amazingly efficient: On an HP9000r735, a com-Ž . plete enumeration of, e.g., all C -structures of which there are 1812 60 needs about 12 sec, 6.5 sec for the generation of sufficiently many