The graph of the icosahedron, shown to the right, can have its edges coloured with
five colours in such a way no two edges of the same colour meet at any vertex. Such
a colouring is called an "edge-5-colouring". The purpose of these pages is to list
all possible edge-5-colourings of the icosahedral graph.
A remarkably symmetrical edge-5-colouring of the icosahedron
The most symmetrical 5-edge-colouring of the icosahedron
One way to edge-5-colour the icosahedron is as follows. Regard it as embedded
symmetrically in 3-space (you were probably doing that already), and assign the
same colour to two edges iff they are (orthogonal or parallel). This colouring
is shown to the left.
This colouring has some remarkable properties:
Each of the 20 cyclically-ordered
A cyclical ordering of a set can be regarded as an arrangement of its elements in a
circle (rather than a list, as for a standard ordering). Thus (a,b,c), (b,c,a) and
(c,a,b) are all the same cyclic ordering; but (a,c,b) is a different cyclic
ordering. three-element subsets of the set of colours
appears once around one of the 20 faces.
Each of the 12 even permutations
Just as permutations of ordered sets can be classified as "even" (having any number
of odd-length cycles and an even number of even-length cycles) and "odd" (having any
number of odd-length cycles and an odd number of even-length cycles), so can
cyclic permutations of sets of odd size. of the set of five colours
appears around one of the 12 vertices.
The partition of the set of edges into five subsets (one for each colour) is
preserved by rotations and reflections of the icosahedron in 3-space.
Each of the 24 rotations whose axes passes through two vertices does a 5-cycle
on the colours.
Each of the 20 rotations whose axes passes through two face-centes does a 3-cycle
on three colours and fixes the other two.
Each of the 15 rotations whose axes passes through two edge-centres, and each of
the 15 reflections, does two 2-cycles on the colours and fixes the fifth.
Other edge-5-colourings of the icosahedron
There are other edge-5-colourings of the icosahedron, harder to find and less
interesting than the one shown and described above. Ignoring duplicates (by rotation,
reflection, and permutation of the colours) there are 17 others, six with mirror
symmetry and eleven chiral.
This page gives some methods and material to help in finding
them, and this list shows all 18.