Platonic graph

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

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.

• 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

External links

• Weisstein, Eric W. "Platonic graph". MathWorld.