Introduction
Fullerenes are a new form of carbon atoms discovered a few
years ago. In 1985 the first C60 clusters were
produced and it was assumed [16] that its 60 carbon atoms are
placed at the vertices of a truncated icosahedron (which
is commonly called a foot ball or soccer ball.)
The truncated icosahedron is one of the 14 Archimedean
solids already known to the ancient Greeks. Five years later
[15][19] the original hypotheses on
the structure of C60 could be confirmed. In the
meantime fullerenes play an important role in chemistry and many
other forms of fullerenes were detected.
In my talk today I will present some examples how
combinatorics under finite group action can be applied in
order to determine the number of essentially different
colourings of the soccer ball. From the chemical point of view this
means isomer enumeration. Furthermore I will introduce a
multi-dimensional cycle index which represents the simultaneous
action of the symmetry group of a fullerene on its sets of
vertices, edges and faces. Finally I will describe the so called
leapfrog principle, which was invented by Fowler [6][9]. It can be used for the
construction of a fullerene C3v from a fullerene
Cv having the same or even a bigger symmetry
group than Cv.
harald.fripertinger@kfunigraz.ac.at
