A007022 Number of 4-regular polyhedra with n nodes.

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 11, 18, 58, 139, 451, 1326, 4461, 14554, 49957, 171159, 598102, 2098675, 7437910, 26490072, 94944685, 341867921, 1236864842, 4493270976, 16387852863, 59985464681, 220320405895, 811796327750, 3000183106119

Offset: 1

Author

N. J. A. Sloane, Apr 28 1994

Keywords

• nonn
• changed

Comments

Number of simple 4-regular 4-edge-connected 3-connected planar graphs; by Steinitz's theorem, every such graph corresponds to a single planar map up to orientation-reversing isomorphism. Equivalently, number of 3-connected quadrangulations of sphere with orientation-reversing isomorphisms permitted with n faces. - Andrey Zabolotskiy, Aug 22 2017

Examples

			For n=6, the sole 6-vertex 4-regular polyhedron is the octahedron. The corresponding 6-face quadrangulation is its dual graph, i. e., the cube graph.
From _Allan Bickle_, May 13 2024: (Start)
For n=8, the unique graph is the square of an 8-cycle.
For n=9, the unique graph is the dual of the Herschel graph. (End)
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Gunnar Brinkmann, Sam Greenberg, Catherine Greenhill, Brendan D. McKay, Robin Thomas, and Paul Wollan, Generation of simple quadrangulations of the sphere, Discr. Math., 305 (2005), 33-54. doi:10.1016/j.disc.2005.10.005
• Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
• Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
• CombOS - Combinatorial Object Server, generate planar graphs
• Michael B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.
• Slavik V. Jablan, Ljiljana M. Radović, and Radmila Sazdanović, Basic polyhedra in knot theory, Kragujevac J. Math. (2005) Vol. 28, 155-164.
• Jorik Jooken, Computer-assisted graph theory: a survey, arXiv:2508.20825 [math.CO], 2025. See Ref. 197 at p. 5.
• T. Tarnai, F. Kovács, P. W. Fowler, and S. D. Guest, Wrapping the cube and other polyhedra, Proc. Roy. Soc. A 468(2145) (2012), 2652-2666. DOI: 10.1098/rspa.2012.0116.

Crossrefs

Cf. A000944 (all polyhedral graphs), A113204, A078672, A078666 (total number of simple 4-regular 4-edge-connected planar maps, including not 3-connected ones).

Cf. A072552, A078666, A111361, A292515 (4-regular planar graphs with restrictions).

Extensions

More terms from Hugo Pfoertner, Mar 22 2003

a(29) corrected by Brendan McKay, Jun 22 2006

Leading zeros prepended by Max Alekseyev, Sep 12 2016

Offset corrected by Andrey Zabolotskiy, Aug 22 2017