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.

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3-space, Hadwiger’s conjecture on t-colorability of graphs with no Kt+1 minor, Tutte’s edge 3-coloring conjecture on edge 3-colorability of 2-connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2-colorability of hypergraphs, and sign-nonsingular matrices.

We discuss splitter theorems for internally 4-connected graphs and for cyclically 5-connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3-coloring conjecture, and Pfaffian orientations of bipartite graphs.

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 this article we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3-connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum degree 5. Key words: planar triangulation, cubic graph, generation, fullerene

Tensegrities are unique, space-filling structures consisting of disjoint rigid elements (rods) connected by tensile elements (strings), which hold their shape due to a synergistic balance of opposing forces. Due to their complexity there are few effective analytical methods for discovering new, and particularly, large irregularly shaped tensegrity structures. Recent efforts using evolutionary search have been moderately successful, but have relied upon a direct encoding of the structure, and therefore face scalability issues [3]. By contrast we employ to a developmental representation grammatically “grow ” tensegrity structures, and as such, issues of scalability, both in terms of representation and of performance, are addressed. Specifically we evolve map L-systems [4] which produce planar graphs [1, 2] corresponding to the structural