Cubic Polyhedral Graph

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

The following table summarizes some named cubic 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 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).

There exists a cubic polyhedral platypus graph of order iff is even and (Goedgebeur et al. 2020).