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 chromatic-index-critical graph G on n vertices is non-trivial if it has at most \Deltab n 2 c edges. We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic index critical graphs on 11 vertices. Together with known results this implies that there are precisely three non-trivial chromaticindex -critical graphs of order 12. 1 Introduction A famous theorem of Vizing [20] states that the chromatic index Ø 0 (G) of a simple graph G is \Delta(G) or \Delta(G) + 1, where \Delta(G) denotes the maximum vertex degree in G. A graph G is class 1 if Ø 0 (G) = \Delta(G) and it is class 2 otherwise. A class 2 graph G is (chromatic index) critical if Ø 0 (G \Gamma e) ! Ø 0 (G) for each edge e of G. If we want to stress the maximum vertex degree of a critical graph G we say G is \Delta(G)-critical. Critical graphs of odd order are easy to construct while not much is known about critical graphs of even order. One reas...

The generation of discrete structures up to isomorphism is interesting as well for theoretical as for practical purposes. Mathematicians want to look at and analyse structures and for example chemical industry uses mathematical generators of isomers for structure elucidation. The example chosen in this paper for explaining general generation methods is a relatively far reaching and fast graph generator which should serve as a basis for the next more powerful version of MOLGEN, our generator of chemical isomers. 1

. Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions. 1. A General Point of View A natural goal in mathematical theories is a full description of the objects that are investigated. This goal has been successfully achieved in some cases, for example all finite abelian groups and with much more effort all finite simple groups. More often one restricted the research activity firstly to more modest problems like the pure existence of any object with som...

In this note we give the smallest 4-regular 4-chromatic graphs with girth 5. There are exactly one graph on 21 vertices and one on 25 vertices. Starting with a conjecture of Grunbaum [2] saying that (k; k; g)-graphs (that is k-chromatic k-regular graphs with girth at least g) exist for all k 2 and g 4 a lot of work has been done investigating such graphs. In [4] Song Wenjie et.al. show that (4; 4; 4) graphs of order n exist if and only if n ?= 12. Furthermore they define f(4; 4; g) to be the smallest number of vertices so that a (4; 4; g) graph exists and conjecture that f(4; 4; 5) = 25. This conjecture is false. In [1] the first author lists the (4; 4; 5) graph on 21 vertives given in figure one. The graph was constructed with the help of a computer, and it was also shown that it is the smallest possible (4; 4; 5) graph. So we have f(4; 4; 5) = 21. A further result was that there are no (4; 4; 5) graphs on 22 or 23 vertices. Using his completely independent generation program genreg...