Platonic graph

Graph with a Platonic solid as its skeleton
Orthographic projections and Schlegel diagrams with Hamiltonian cycles of the vertices of the five platonic solids – only the octahedron has an Eulerian path or cycle, by extending its path with the dotted one
  • v
  • t
  • e
The platonic graphs can be seen as Schlegel diagrams of the platonic solids. (excluding the square pyramid also shown here)

In the mathematical field of graph theory, a Platonic graph is a graph that has one of the Platonic solids as its skeleton. There are 5 Platonic graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-vertex-connected, vertex-transitive, edge-transitive and planar graphs), and also Hamiltonian graphs.[1]

  • Tetrahedral graph – 4 vertices, 6 edges
  • Octahedral graph – 6 vertices, 12 edges
  • Cubical graph – 8 vertices, 12 edges
  • Icosahedral graph – 12 vertices, 30 edges
  • Dodecahedral graph – 20 vertices, 30 edges
Orthogonal projections of platonic solids

See also

  • Regular map (graph theory)
  • Archimedean graph
  • Wheel graph

References

  1. ^ Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 266.

External links


Stub icon

This graph theory-related article is a stub. You can help Wikipedia by expanding it.

  • v
  • t
  • e