In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a top-down approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGI-workstation. The method described can easily be generalised for 3-regular spherical maps with no face having more than 6 edges in its boundary.

We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp and we present all nonhamiltonian such graphs of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vertices. That all 3-connected cubic planar graphs on at most 176 vertices and with face size at most 6 are hamiltonian is also verified. 1 Introduction. In this paper we describe an investigation (making much use of computation) of cyclically k-connected cubic planar graphs (CkCPs) for k = 4, 5 and report the results. We shall also have occasion to consider cubic 3-connected planar graphs with no restriction on cyclic connectivity; these we refer to as C3CPs. The investigation

In the present paper a method for generating fullerenes at random is presented. It is based on the well known Stone-Wales (SW) transformation. The method could be further generalised so that other trivalent polyhedra with prescribed properties are generated. 1. INTRODUCTION Fullerenes and other pure carbon cages remain a subject of rigorous research. The mechanism of fullerenes growth is still not fully understood although much has been learned 1 . In the present paper a method for generating fullerenes at random is presented. It is based on the well known Stone-Wales (SW) transformation 2; 3 and it has been successfully implemented as a part of the VEGA: a system for manipulating discrete mathematical structures. 4; 5 The method could be further generalised so that it is able to narrow the population of carbon cages with special properties. From a mathematical standpoint a fullerene is planar trivalent graph whose faces are pentagons and hexagons. It turns out that the number...