Colourings of the fullerene C60

## Colourings of the fullerene C60

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 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 different colourings using k colours via Pólya-theory by replacing each variable xi in the cycle index by k. In SYMMETRICA there is the routine `polya_const_sub(a,b,c)` which does this substitution. `a` is the cycle index, `b` takes the constant k and `c` is the result. The numbers of 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

harald.fripertinger@kfunigraz.ac.at,
last changed: January 23, 2001

 Colourings of the fullerene C60