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A081621 Number of n-node triangulations of the sphere with minimal degree 5. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,13

COMMENTS

Other face sizes bigger 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.

REFERENCES

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

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

LINKS

Table of n, a(n) for n=4..40.

Gunnar Brinkmann and Brendan McKay, Plantri and fullgen programs for generation of certain types of planar graph.

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

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

Hugo Pfoertner, Icosahedral best coverings.

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 A084921 A070765

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

STATUS

approved

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Last modified April 20 22:29 EDT 2014. Contains 240823 sequences.