A081621 Number of n-node triangulations of the sphere with minimal degree 5.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 4, 12, 23, 73, 192, 651, 2070, 7290, 25381, 91441, 329824, 1204737, 4412031, 16248772, 59995535, 222231424, 825028656, 3069993552, 11446245342, 42758608761, 160012226334, 599822851579, 2252137171764, 8469193859271, 31896058068930

Offset

4,13

Comments

Other face sizes larger than 5 and 6 are allowed and there can be more than 12 vertices with degree 5.

Convex polytopes with minimum degree at least 5. The sequence is extracted from the file more-counts.txt that comes with the plantri distribution.

Grace conjectured that all polyhedra inscribed in the unit sphere with maximal volume are "medial" (all faces triangular and vertex degree either m or m+1 where m < 6 - 12/n < m+1). For n = 12 and n > 13 the medial polyhedra have 12 vertices of degree 5 and n-12 vertices of degree 6. All known numerical solutions of the maximal volume problem (A081314) have this property.

The triangulated arrangements of points on a sphere with icosahedral symmetry given by Hardin, Sloane and Smith are examples for large n.

Links

Table of n, a(n) for n=4..41.

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]

G. Brinkmann and B. D. McKay, Construction of planar triangulations with minimum degree 5, Discr. Math. 301 (2005), 147-163.

CombOS - Combinatorial Object Server, generate planar graphs.

D. W. Grace, Search for largest polyhedra, Math. Comp. 17, 197-199 (1963).

R. H. Hardin, N. J. A. Sloane and W. D. Smith, Spherical Codes with Icosahedral Symmetry.

Hugo Pfoertner, Icosahedral best coverings.

Sage, Common Graphs.

Thom Sulanke, Generating triangulations of surfaces (surftri), (also subpages).

Example

With vertices denoted by letters a, b, ... the neighbor lists are for a(14)=1: (bcdef, afghc, abhid, acije, adjkf, aeklgb, bflmh, bgmic, chmnjd, dinke, ejnlf, fknmg, glnih, imlkj).
a(15)=1: (bcdefg, aghic, abijd, acjke, adklf, aelmg, afmhb, bgmni, bhnjc, cinokd, djole, ekomf, flonhg, hmoji, jnmlk); a(16)=3: (bcdef, afghc, abhijd, acjke, adklf, aelmgb, bfmnh, bgnic, chnoj, ciokd, djople, ekpmf, flpng, gmpoih, inpkj, konml), (bcdef, afghc, abhijd, acjke, adklf, aelmgb, bfmnh, bgnic, chnoj, ciopkd, djple, ekpmf, flpong, gmoih, inmpj, jomlk), (bcdef, afghijc, abjkd, ackle, adlmf, aemgb, bfmnh, bgnoi, bhopj, bipkc, cjpld, dkponme, elngf, gmloh, hnlpi, iolkj).

Crossrefs

Cf. A000109, A000103, A081314.

Sequence in context: A006791 A111358 A111357 * A073713 A291023 A084921

Adjacent sequences: A081618 A081619 A081620 * A081622 A081623 A081624

Keyword

nonn

Author

Hugo Pfoertner, Mar 24 2003

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

a(41) computed with plantri by Jan Goedgebeur, Dec 03 2021

Status

approved