A colouring problem for the dodecahedral graph
| Autor: | Tibor Tarnai, Endre Makai |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Coxeter group
Astrophysics::Cosmology and Extragalactic Astrophysics Computer Science::Computational Geometry Vertex (geometry) Combinatorics Dodecahedron Congruence (geometry) Tetrahedron Mathematics::Metric Geometry Graph (abstract data type) Ball (mathematics) Astrophysics::Galaxy Astrophysics Inscribed figure Mathematics |
| Zdroj: | Elemente der Mathematik. 76:1-9 |
| ISSN: | 0013-6018 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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