# A colouring problem for the dodecahedral graph

## TL;DR

This paper investigates vertex colourings of the dodecahedral graph with five colours, showing there are exactly 240 such colourings related to the dodecahedron's symmetries and associated tetrahedral compounds.

## Contribution

It provides a complete enumeration and classification of five-colourings of the dodecahedral graph with geometric and combinatorial proofs, linking colourings to tetrahedral compounds.

## Key findings

- Total of 240 colourings exist.
- Colour classes form vertices of five tetrahedra.
- Colourings relate to tetrahedral compounds within the dodecahedron.

## Abstract

We consider vertex colourings of the dodecahedral graph with five colours, such that on each face the vertices are coloured with all the five colours. We show that the total number of these colourings is 240. All such colourings can be obtained from any given such colouring, by permuting the colours, and possibly applying central symmetry with respect to the centre of the regular dodecahedron. For any such colouring, the colour classes form the vertex sets of five regular tetrahedra. These tetrahedra together form one of the two compounds of five tetrahedra, inscribed in the regular dodecahedron. We give two proofs: a combinatorial one, and a geometrical one. Our result is related to the result in W. W. Rouse Ball -- H. S. M. Coxeter, stating that there are four such colourings, as follows. There are four such colourings, up to applying an arbitrary orientation-preserving congruence of the regular dodecahedron.