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A081621 Number of n-node triangulations of the sphere with minimal degree 5. 4


%S 0,0,0,0,0,0,0,0,1,0,1,1,3,4,12,23,73,192,651,2070,7290,25381,91441,

%T 329824,1204737,4412031,16248772,59995535,222231424,825028656,

%U 3069993552,11446245342,42758608761,160012226334,599822851579,2252137171764,8469193859271

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

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

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

%C 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.

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

%H Gunnar Brinkmann and Brendan McKay, <a href="http://users.cecs.anu.edu.au/~bdm/plantri/">plantri and fullgen</a> programs for generation of certain types of planar graph.

%H Gunnar Brinkmann and Brendan McKay, <a href="/A000103/a000103_1.pdf">plantri and fullgen</a> programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]

%H G. Brinkmann and B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/plantri/min5paper_sc.pdf">Construction of planar triangulations with minimum degree 5</a>, Discr. Math. 301 (2005), 147-163.

%H CombOS - Combinatorial Object Server, <a href="http://combos.org/plantri">generate planar graphs</a>

%H D. W. Grace, <a href="https://doi.org/10.1090/S0025-5718-63-99183-X">Search for largest polyhedra.</a> Math. Comp. 17, 197-199 (1963)

%H R. H. Hardin, N. J. A. Sloane and W. D. Smith, <a href="http://neilsloane.com/icosahedral.codes/">Spherical Codes with Icosahedral Symmetry.</a>

%H Hugo Pfoertner, <a href="http://www.enginemonitoring.org/sphere/icoscov.pdf">Icosahedral best coverings.</a>

%H Sage, <a href="http://doc.sagemath.org/html/en/reference/graphs/sage/graphs/graph_generators.html">Common Graphs.</a>

%H Thom Sulanke, <a href="http://hep.physics.indiana.edu/~tsulanke/graphs/surftri/">Generating triangulations of surfaces (surftri)</a>, (also subpages).

%e 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)

%e 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);

%Y Cf. A000109, A000103, A081314.

%K nonn

%O 4,13

%A _Hugo Pfoertner_, Mar 24 2003

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

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