A colouring of the vertices, edges or faces of a fullerene with
k colours can be interpreted as a function from the set of
all vertices, edges or faces into the set of k colours. Two
colourings are called essentially different if and only if
the corresponding functions lie in different orbits of the group
R or S acting on the set of all these functions in a
natural way. This means that the group is acting on the domain of
these functions. From the cycle indices above you can compute the
number of essentially different colourings using k
colours via Pólya-theory by replacing each variable
xi in the cycle index by k. The numbers of
essentially different colourings with 2 colours are given
in table.
R
S
Vertices
19215.358678.900736
9607.679885.269312
Edges
20.632333.988107.263792.381952
10.316166.994124.293843.474944
Faces
71.600640
35.931952
Different colourings of the fullerene
C60 with 2 colours
