# Edge-5-colourings of the Icosahedron

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

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 three-element subsets of the set of colours appears once around one of the 20 faces.
• Each of the 12 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.

This is one of several miscellaneous pages.