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.
Tabulations of cubic polyhedral graphs are commonly limited to those that are triangle-free (e.g., Read and Wilson 1998). For
,
2, ..., the numbers of triangle-free cubic
polyhedral graphs on 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).
Bowen, R. and Fisk, S. "Generation of Triangulations of the Sphere." Math. Comput.21, 250-252, 1967.Goedgebeur,
J.; Neyt, A.; and Zamfirescu, C. T. "Structural and Computational Results
on Platypus Graphs." Appl. Math. Comput., 386:125491, 10 pages, 2020.House
of Graphs. Cubic Polyhedral Graphs. Tetrahedron
K_4, The cube Q_3, Tutte
16-graph, and others.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."
,
4, ... nodes are 0, 1, 1, 2, 5, 14, 50, 233, 1249,
... (OEIS A000109).
-
)
-
)
,
2, ..., the numbers of triangle-free cubic
polyhedral graphs on 
nodes are 0, 0, 0, 1, 1,
2, 5, 12, 34, ... (OEIS A000103).
iff 
is even and 
(Goedgebeur et al. 2020).
