Gunnar Brinkmann

A Constructive Enumeration of Fullerenes

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 ofC60in less than 20 s on an SGI-workstation. The method described can easily be generalized for 3-regular spherical maps with no face having more than 6 edges in its boundary.

House of Graphs: A database of interesting graphs

In this note we present House of Graphs (http://hog.grinvin.org) which is a new database of graphs. The key principle is to have a searchable database and offer-next to complete lists of some graph classes-also a list of special graphs that have already turned out to be interesting and relevant in the study of graph theoretic problems or as counterexamples to conjectures. This list can be extended by users of the database.

A census of nanotube caps

Abstract The number of hemi-fullerene caps compatible with a carbon nanotube of given diameter and helicity is determined for all tubes up to a diameter of 30 A. This census should prove useful as a basis for future systematic studies of the chemistry and physics of capped nanotubes.

Fast generation of cubic graphs

In this paper an efficient algorithm to generate regular graphs with small vertex valency is presented. The running times of a program based on this algorithm and designed to generate cubic graphs are below two natural benchmarks: (a) If N(n) denotes the number of pairwise non-isomorphic cubic graphs with n vertices and T(n) the time needed for generating the list of all these graphs, then T(n)/N(n) decreases gradually for the observed values of n. This suggests that T(n)/N(n) might be uniformly bounded for all n, ignoring the time to write the outputs, but we are unable to prove this and in fact are not confident about it. (b) For programs that generate lists of non-isomorphic objects, but cannot a priori make sure to avoid the generation of isomorphic copies, the time needed to check a randomly ordered list of these objects for being non-isomorphic is a natural benchmark. Since for large lists (n ≥ 22, girth 3) existing graph isomorphism programs take longer to canonically label all of the N(n) graphs than our algorithm takes to generate them, our algorithm is probably faster than any method which does one or more isomorphism test for every graph.

Generation and properties of snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for snarks, the class of non-trivial 3-regular graphs which cannot be 3-e ...

Posets on up to 16 Points

In this article we describe a very efficient method to construct pairwise non-isomorphic posets (equivalently, T0 topologies). We also give the results obtained by a computer program based on this algorithm, in particular the numbers of non-isomorphic posets on 15 and 16 points and the numbers of labelled posets and topologies on 17 and 18 points.

Generation of simple quadrangulations of the sphere

A simple quadrangulation of the sphere is a finite simple graph embedded on the sphere such that every face is bounded by a walk of 4 edges. We consider the following classes of simple quadrangulations: arbitrary, minimum degree 3, 3-connected, and 3-connected without non-facial 4-cycles. In each case, we show how the class can be generated by starting with some basic graphs in the class and applying a sequence of local modifications. The duals of our algorithms generate classes of quartic (4-regular) planar graphs. In the case of minimum degree 3, our result is a strengthening of a theorem of Nakamoto and almost implicit in Nakamotos proof. In the case of 3-connectivity, a corollary of our theorem matches a theorem of Batagelj. However, Batageljs proof contained a serious error which cannot easily be corrected. We also give a theoretical enumeration of rooted planar quadrangulations of minimum degree 3, and some counts obtained by a program of Brinkmann and McKay that implements our algorithm.

Counting symmetric configurations v 3

Abstract In this article we give tables of configurations v 3 for v ⩽18 and triangle-free configurations for v ⩽21 together with some statistics about some properties of the structures like transitivity, self-duality or self-polarity.

The Smallest Cubic Graphs of Girth Nine

We describe two computational methods for the construction of cubic graphs with given girth. These were used to produce two independent proofs that the (3, 9)-cages, defined as the smallest cubic graphs of girth 9, have 58 vertices. There are exactly 18 such graphs. We also show that cubic graphs of girth 11 must have at least 106 vertices and cubic graphs of girth 13 must have at least 196 vertices.

The generation of fullerenes.

We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes - fullgen - and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are Hamiltonian and verify that the smallest counterexample to the spiral conjecture has 380 vertices.