A000103 -id:A000103 - cp's OEIS frontend

A005964 Number of trivalent connected (or cubic) planar graphs with 2n nodes.

Original entry on oeis.org

0, 1, 1, 3, 9, 32, 133, 681, 3893, 24809, 169206, 1214462, 9034509, 69093299, 539991437

Offset: 1

N. J. A. Sloane

A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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]
F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 3.
M. Meringer, Tables of Regular Graphs.
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, Cubic Graph.

Cf. A058378, A000109, A002851, A204186.

Extended by Brendan McKay and Gunnar Brinkmann using their program "plantri", Dec 19 2000

A002880 Number of 3-connected nets with n edges.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 9, 11, 37, 79, 249, 671, 2182, 6692, 22131, 72405, 243806, 822788, 2815119, 9679205, 33551192, 116900081, 409675567, 1442454215, 5102542680, 18124571838, 64634480340, 231334873091, 830828150081, 2993489821771

Offset: 6

N. J. A. Sloane

Also, the number of 3-connected quadrangulations without separating 4-cycles (up to orientation) with n faces. - Andrey Zabolotskiy, Sep 20 2019

			G.f. = x^6 + x^8 + x^9 + 2*x^10 + 2*x^11 + 9*x^12 + 11*x^13 + 37*x^14 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
J. A. D. Cameron, Searching for Squared Squares, USYD Master of Science Thesis (1976). Rare Books & Special Collections Fisher Library, Sydney University.
J. A. D. Cameron, Table 7.2 - listing of tri-connected planar graphs by edge, from the thesis - this was the first count of order 20 (22131). Photo by Stuart E Anderson.
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.
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.
P. J. Federico, Enumeration of polyhedra: the number of 9-hedra, J. Combin. Theory, 7 (1969), 155-161.
N. D. Kazarinoff and R. Weitzenkamp, Squaring rectangles and squares, Amer. Math. Monthly, 80 (1973), 877-888.

Cf. A002840, A002841, A113201, A078666, A007022, A113205, A338511.

A113201 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms permitted.

Original entry on oeis.org

1, 1, 2, 3, 9, 18, 62, 198, 803, 3378, 15882, 77185, 393075, 2049974, 10938182, 59312272, 326258544, 1815910231, 10213424233, 57974895671, 331820721234, 1913429250439, 11109119321058, 64901418126997

Offset: 4

N. J. A. Sloane, Jan 07 2006

nonn

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.
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]
J. Cantarella, H. Chapman, M. Mastin, Knot Probabilities in Random Diagrams, arXiv preprint arXiv:1512.05749 [math.GT], 2015. See Tables 1.
CombOS - Combinatorial Object Server, generate planar graphs
Paul Jungeblut, Edge Guarding Plane Graphs, Master Thesis, Karlsruhe Institute of Technology (Germany, 2019).
Sage, Common Graphs

Cf. A078666, A007022, A002880.

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

Original entry on oeis.org

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

Hugo Pfoertner, Mar 24 2003

nonn

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.

			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).

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).

Cf. A000109, A000103, A081314.

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

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

A111358 Numbers of planar triangulations with minimum degree 5 and without separating 3- or 4-cycles - that is 3- or 4-cycles where the interior and exterior contain at least one vertex.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 12, 23, 71, 187, 627, 1970, 6833, 23384, 82625, 292164, 1045329, 3750277, 13532724, 48977625, 177919099, 648145255, 2368046117, 8674199554, 31854078139, 117252592450, 432576302286, 1599320144703, 5925181102878

Offset: 12

Gunnar Brinkmann, Nov 07 2005

nonn

A006791 and this sequence are the same sequence. The correspondence is just that these objects are planar duals of each other. But the offset and step are different: if the cubic graph has 2*n vertices, the dual triangulation has n+2 vertices. - Brendan McKay, May 24 2017

Also the number of 5-connected triangulations on n vertices. - Manfred Scheucher, Mar 17 2023

			The icosahedron is the smallest triangulation with minimum degree 5 and it doesn't contain any separating 3- or 4-cycles. Examples can easily be seen as 2D and 3D pictures using the program CaGe cited above.

G. Brinkmann, CaGe.
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 Brendan D. McKay, Construction of planar triangulations with minimum degree 5 , Disc. Math. vol 301, iss. 2-3 (2005) 147-163.
D. A. Holton and B. D. McKay, The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices, J. Combinat. Theory B vol 45, iss. 3 (1988) 305-319.
D. A. Holton and B. D. McKay, Erratum, J. Combinat. Theory B vol 47, iss. 2 (1989) 248.

Cf. A081621, A007894, A006791, A000109, A007021, A361578.

A113204 Same as A007022, but with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 4, 3, 15, 25, 92, 234, 803, 2469, 8512, 28290, 98148, 338673, 1188338, 4180854, 14840031, 52904562, 189724510, 683384218, 2472961423, 8984888982, 32772085447, 119963084542, 440623586740, 1623555117611

Offset: 1

N. J. A. Sloane, Jan 07 2006

nonn

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

Cf. A113201, A078666, A007022, A002880, A113202, A113205.

a(29) corrected by Brendan McKay, Jun 22 2006
Leading zeros prepended by Max Alekseyev, Sep 13 2016
Offset corrected by Andrey Zabolotskiy, Feb 09 2018

A113205 Same as A002880, but with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 12, 16, 59, 133, 445, 1248, 4162, 13014, 43474, 143304, 484444, 1639388, 5617205, 19332596, 67048051, 233691112, 819121608, 2884443024, 10204104900, 36247138920, 129264732757, 462661038926, 1661637913984

Offset: 6

N. J. A. Sloane, Jan 07 2006

nonn

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.
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

Cf. A113201, A078666, A007022, A002880, A113202, A113203, A113204.

A058378 Number of trivalent 2-connected planar graphs with 2n nodes.

Original entry on oeis.org

0, 1, 1, 3, 8, 29, 114, 583, 3310, 21168, 144622, 1039495, 7731540

Offset: 1

N. J. A. Sloane, Dec 19 2000

A. T. Balaban, Enumeration of Cyclic Graphs, pp. 63-105 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 92.
Computed by Brendan McKay and Gunnar Brinkmann using their program "plantri", Dec 19 2000.

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]
Gunnar Brinkmann and Brendan McKay, Guide to using plantri [Cached copy, with permission]
M. Meringer, Tables of Regular Graphs

Cf. A005964, A000109, A387146.

A113202 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

1, 1, 2, 3, 10, 21, 83, 298, 1339, 6049, 29765, 148842, 770267, 4054539, 21743705, 118237471, 651370528, 3628421181, 20416662314, 115919209155, 663548898942, 3826577783917, 22217382001865, 129800215435088

Offset: 4

N. J. A. Sloane, Jan 07 2006

nonn

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.
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

Cf. A113201, A078666, A007022, A002880, A113203, A113204, A113205.

A113203 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, minimal degree 3, with orientation-reversing isomorphisms forbidden.

Original entry on oeis.org

1, 0, 1, 1, 4, 3, 16, 26, 99, 256, 895, 2789, 9740, 32799, 115024, 401180, 1421170, 5046539, 18066772, 64940825, 234712099, 851801048, 3104690139, 11358900851, 41710948878, 153684688127, 568079430741, 2106188450292

Offset: 8

N. J. A. Sloane, Jan 07 2006

nonn

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.
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

Cf. A113201, A078666, A007022, A002880, A113202, A113204, A113205.

cp's OEIS Frontend

A005964 Number of trivalent connected (or cubic) planar graphs with 2n nodes.

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A002880 Number of 3-connected nets with n edges.

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A113201 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms permitted.

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

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A111358 Numbers of planar triangulations with minimum degree 5 and without separating 3- or 4-cycles - that is 3- or 4-cycles where the interior and exterior contain at least one vertex.

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A113204 Same as A007022, but with orientation-reversing isomorphisms forbidden.

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A113205 Same as A002880, but with orientation-reversing isomorphisms forbidden.

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A058378 Number of trivalent 2-connected planar graphs with 2n nodes.

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A113202 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, with orientation-reversing isomorphisms forbidden.

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A113203 Number of isomorphism classes of simple quadrangulations of the sphere having n vertices and n-2 faces, minimal degree 3, with orientation-reversing isomorphisms forbidden.

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