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# Cubic Polyhedral Graph

A cubic polyhedral graph is a graph that is both cubic and polyhedral. The numbers of cubical polyhedral graphs on , 4, ... nodes are 0, 1, 1, 2, 5, 14, 50, 233, 1249, ... (OEIS A000109).

The following table summarizes some named cubical polyhedral graphs and classes of graphs, some of which are illustrated above.

 -prism graph -generalized Petersen graph 4 tetrahedral graph 8 cubical graph 12 Frucht graph 12 truncated tetrahedral graph 18 truncated prism graph 20 dodecahedral graph 24 truncated cubical graph 24 truncated octahedral graph 38 Barnette-Bosák-Lederberg graph 42 Faulkner-Younger graph 42 42 Grinberg graph 42 44 Faulkner-Younger graph 44 44 Grinberg graph 44 46 Grinberg graph 46 46 Tutte's graph 48 great rhombicuboctahedral graph 60 truncated dodecahedral graph 60 truncated icosahedral graph 94 Thomassen graph 94 120 great rhombicosidodecahedral graph 124 Grünbaum graph 124

Tabulations of cubic polyhedral graphs are commonly limited to those that are triangle-free (e.g., Read and Wilson 1998). The numbers of -node triangle-free cubic polyhedral graphs on , 2, ... nodes are 0, 0, 0, 1, 1, 2, 5, 12, 34, (OEIS A000103).

The figure above illustrates triangle-free cubic polyhedral graphs up to 18 vertices, together with their notation in Read and Wilson (1998).

Cubic Graph, Cubic Symmetric Graph, Polyhedral Graph, Simple Polyhedron

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## References

Bowen, R. and Fisk, S. "Generation of Triangulations of the Sphere." Math. Comput. 21, 250-252, 1967.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, 1998.Sloane, N. J. A. Sequences A000103/M1423 and A000109/M1469 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Cubic Polyhedral Graph

## Cite this as:

Weisstein, Eric W. "Cubic Polyhedral Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubicPolyhedralGraph.html