Platonic Graph

A polyhedral graph corresponding to the skeleton of a Platonic solid. The five platonic graphs, the tetrahedral graph, cubical graph, octahedral graph, dodecahedral graph, and icosahedral graph, are illustrated above. They are special cases of Schlegel graphs.

Platonic graphs are graceful (Gardner 1983, pp. 158 and 163-164).

The following table summarizes the Platonic graphs and some of their properties.

graph	regularity				Hamiltonian	Eulerian	vertex-transitive	edge-transitive
cubical graph	cubic	8	12	48	yes	no	yes	yes
dodecahedral graph	cubic	20	30	120	yes	no	yes	yes
icosahedral graph	quintic	12	30	120	yes	no	yes	yes
octahedral graph	quartic	6	12	48	yes	yes	yes	yes
tetrahedral graph	cubic	4	6	24	yes	no	yes	yes

REFERENCES

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 234, 1976.

Gardner, M. "Golomb's Graceful Graphs." Ch. 15 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 152-165, 1983.

Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, pp. 263 and 266, 1998.