A007022 - OEIS

A007022

Number of 4-regular polyhedra with n nodes.
(Formerly M2290)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,10

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..34.

G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas, and P. 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

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory Series B 66:1 (1996), 87-122.

S. V. Jablan, L. M. Radović, and R. Sazdanović, Basic polyhedra in knot theory, Kragujevac J. Math., 28 (2005), 155-164.

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.

EXAMPLE

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)

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).

Sequence in context: A347521 A278835 A163932 * A011950 A343465 A289883

Adjacent sequences: A007019 A007020 A007021 * A007023 A007024 A007025

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Apr 28 1994

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

STATUS

approved