A111361 The number of 4-regular plane graphs with n vertices with all faces 3-gons or 4-gons.

0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 5, 2, 8, 5, 12, 8, 25, 13, 30, 23, 51, 33, 76, 51, 109, 78, 144, 106, 218, 150, 274, 212, 382, 279, 499, 366, 650, 493, 815, 623, 1083, 800, 1305, 1020, 1653, 1261, 2045, 1554, 2505, 1946, 3008, 2322, 3713, 2829, 4354, 3418, 5233, 4063, 6234

Offset: 2

Author

Gunnar Brinkmann, Nov 07 2005

Keywords

• nonn

Comments

These are the 4-regular graphs corresponding to the 3-regular fullerenes. Only the two smallest possible face sizes are allowed. The numbers up to a(33) have been checked by 2 independent programs. Further numbers have not been checked independently.

Examples

			From _Allan Bickle_, May 13 2024: (Start)
The smallest example (n=6) is the octahedron (only 3-gons).
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)
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 2..131 (from Hasheminezhad & McKay)
• G. Brinkmann, O. Heidemeier and T. Harmuth, The construction of cubic and quartic planar maps with prescribed face degrees, Discrete Applied Mathematics 128: 541-554, (2003).
• 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]
• Michel-Marie Deza, Mathieu Dutour Sikiric and Mikhail Ivanovitch Shtogrin, Geometric Structure of Chemistry-Relevant Graphs, Springer, 2015; see Sec. 4.4.
• Mathieu Dutour Sikiric and Michel Deza, 4-regular and self-dual analogs of fullerenes, arXiv:0910.5323 [math.GT], 2009.
• Mahdieh Hasheminezhad and Brendan D. McKay, Recursive generation of simple planar quadrangulations with vertices of degree 3 and 4, Discussiones Mathematicae Graph Theory, 30 (2010), 123-136.
• 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. A007894.

Cf. A167156, A167157, A167158, A167159.

Cf. A007022, A072552, A078666, A292515 (4-regular planar graphs with restrictions).

Extensions

Leading zeros prepended, terms a(34) and beyond added from the book by Deza et al. (except for a(60) from the paper by Brinkmann et al.) by Andrey Zabolotskiy, Oct 09 2021