A000109 -id:A000109 - cp's OEIS frontend

A007030 Non-Hamiltonian simplicial polyhedra with n nodes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 30, 239, 2369, 22039, 205663, 1879665, 16999932, 152227187, 1353996482

Offset: 1

a(18) = 1879665 was conjectured by Dillencourt and verified by direct computation by Sean A. Irvine, Sep 26 2017.

By Steinitz's theorem non-Hamiltonian simplicial polyhedra correspond to non-Hamiltonian maximal planar graphs. - William P. Orrick, Feb 25 2021

			The unique non-Hamiltonian maximal planar graph of 11 vertices is the Goldner-Harary graph. A corresponding simplicial polyhedron can be obtained by attaching a tetrahedron to each of the six faces of a triangular bipyramid. - _William P. Orrick_, Feb 25 2021

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory, Series B, Volume 66, Issue 1, January 1996, Pages 87-122.
Eric Weisstein's World of Mathematics, Polyhedral Graph
Wikimedia, Goldner-Harary graphs, additional images of the graph and related simplicial polyhedron created by David Eppstein and Richard J. Mathar. - _William P. Orrick_, Feb 25 2021
Wikipedia, Goldner-Harary graph

Cf. A000109, A115340.

a(n) = A000109(n) - A115340(n-2). - William P. Orrick, Feb 20 2021

a(18) from Sean A. Irvine, Sep 26 2017

a(19)-a(21) using new formula by William P. Orrick, Feb 20 2021

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.

A115340 Number of dual Hamiltonian cubic polyhedra or planar 3-connected Yutsis graphs on 2n nodes.

Original entry on oeis.org

1, 1, 2, 5, 14, 50, 233, 1248, 7593, 49536, 339483, 2404472, 17468202, 129459090, 975647292, 7458907217, 57744122366, 452028275567, 3573870490382

Offset: 2

Dries Van Dyck (VanDyck.Dries(AT)gmail.com), Mar 06 2006

Also, a(n) is the number of Hamiltonian planar triangulations with n+2 vertices. - Brendan McKay, Feb 20 2021

Yutsis graphs are connected cubic graphs which can be partitioned into two vertex-induced trees, which are necessarily of the same size. The cut separating both trees contains n+2 edges for a graph on 2n nodes, forming a Hamiltonian cycle in the planar dual if the graph is planar. These graphs are maximal in the number of nodes of the largest vertex-induced forests among the connected cubic graphs (floor((6n-2)/4) for a graph on 2n nodes). Whitney showed in 1931 that proving the 4-color theorem for a planar Yutsis graph implies the theorem for all planar graphs.

F. Jaeger, On vertex induced-forests in cubic graphs, Proceedings 5th Southeastern Conference, Congressus Numerantium (1974) 501-512.

L. H. Kauffman, Map Coloring and the Vector Cross Product, Journal of Combinatorial Theory, Series B, 48 (1990) 145-154.
D. Van Dyck, G. Brinkmann, V. Fack and B. D. McKay, To be or not to be Yutsis: algorithms for the decision problem, Computer Physics Communications 173 (2005) 61-70.
Dries Van Dyck and Veerle Fack, Yutsis project

Cf. A000109, A007030.

A000109 = Cases[Import["https://oeis.org/A000109/b000109.txt", "Table"], {, }][[All, 2]];
A007030 = Cases[Import["https://oeis.org/A007030/b007030.txt", "Table"], {, }][[All, 2]];
a[n_] := A000109[[n]] - A007030[[n+2]];
Table[a[n], {n, 2, 19}] (* Jean-François Alcover, Jul 20 2022 *)

a(n) = A000109(n+2) - A007030(n+2). - William P. Orrick, Feb 20 2021

a(20) from Van Dyck et al. added by Andrey Zabolotskiy, Sep 10 2024

A253882 Number of 3-connected planar triangulations of the sphere with n vertices up to orientation preserving isomorphisms.

Original entry on oeis.org

1, 1, 2, 6, 17, 73, 389, 2274, 14502, 97033, 672781, 4792530, 34911786, 259106122, 1954315346, 14949368524, 115784496932, 906736988527, 7171613842488, 57231089062625, 460428456484557, 3731572377382341, 30447133566946517, 249968326771680542, 2063931874299323140

Offset: 4

Danny Rorabaugh, Feb 27 2015

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..500
CombOS - Combinatorial Object Server, generate planar graphs
Pascal Honvault, Equivalent classes of degree sequences for triangulated polyhedra and their convex realization, Contributions to Disc. Math. (2021) Vol. 16, No. 1, 128-137.
Pascal Honvault, Local geometry of polyhedra, hal-03744217 [math], 2022.
The House of Graphs, Planar graphs

Cf. A000109 (full automorphism group), A000260 (rooted at an edge), A000944, A002709 (with a distinguished face).

a(n)={if(n<3, 0, (2*binomial(4*(n-3)+1, n-3)/((n-2)*(3*n-7))
  + 3*sumdiv(n-2, d, if(d>=2, my(s=(n-2)/d); eulerphi(d)*binomial(4*s,s))/4)
  + if(n%2==1, my(s=(n-3)/2); 3*binomial(4*s,s)*(2*s+1)/(3*s+1))
  + if(n%3==1, my(s=(n-4)/3); 8*binomial(4*s,s)*(4*s+1)/(3*s+1))
  + if(n%3==0, my(s=(n-3)/3); 2*binomial(4*s,s)) )/(6*(n-2)))} \\ Andrew Howroyd, Mar 02 2021

Name clarified and terms a(24) and beyond from Andrew Howroyd, Mar 02 2021

A133236 Number of bipartite planar graphs with 2n nodes and at least one zero eigenvector.

Original entry on oeis.org

0, 1, 0, 1, 11, 8, 70, 613, 1225, 11330, 120628

Offset: 2

N. J. A. Sloane, Oct 14 2007

nonn

Sciriha, I. and Fowler, P.W., Nonbonding Orbitals in Fullerenes: Nuts and Cores in Singular Polyhedral Graphs J. Chem. Inf. Model., 47, 5, 1763 - 1775, 2007.

Cf. A000109, A007083, A133237.

A133237 Number of bipartite planar graphs with 2n nodes and exactly one zero eigenvector.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 67, 322, 1123, 10548, 81127

Offset: 2

N. J. A. Sloane, Oct 14 2007

nonn

Sciriha, I. and Fowler, P.W., Nonbonding Orbitals in Fullerenes: Nuts and Cores in Singular Polyhedral Graphs J. Chem. Inf. Model., 47, 5, 1763 - 1775, 2007.

Cf. A000109, A007083, A133236.

A222318 Number of 4-dimensional simplicial convex polytopes with n nodes.

Original entry on oeis.org

1, 2, 5, 37, 1142, 162004

Offset: 5

N. J. A. Sloane, Feb 16 2013

Firsching, Moritz Realizability and inscribability for simplicial polytopes via nonlinear optimization. Math. Program. 166, No. 1-2 (A), 273-295 (2017). Table 1
Moritz Firsching, The complete enumeration of 4-polytopes and 3-spheres with nine vertices, Israel Journal of Mathematics 240, 417-441 (2020); arXiv:1803.05205 [math.MG], 2018.
Komei Fukuda, Hiroyuki Miyata and Sonoko Moriyama, Complete Enumeration of Small Realizable Oriented Matroids. Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. Also arXiv:1204.0645 [math.CO], 2012.

Cf. A000109, A005841.

Changed "4" to "n" in description of sequence. - Moritz Firsching, Apr 02 2015

a(10) from M. Firsching's paper added by Andrey Zabolotskiy, Jun 28 2022

A342971 Non-1-tough simplicial polyhedra with n nodes.

Original entry on oeis.org

1, 2, 29, 233, 2297, 21192, 195862

Offset: 11

William P. Orrick, Apr 01 2021

A graph is 1-tough if there is no set of k vertices whose deletion splits the graph into more than k components.

If a graph is not 1-tough then it is not Hamiltonian.

Cf. A000109, A115340, A007030, A007031.

a(n) = A007030(n) - A007031(n).

A007020 Maximal planar degree sequences with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 5, 13, 33, 85, 199, 445, 947, 1909, 3713, 7006, 12765, 22764, 39540

Offset: 3

N. J. A. Sloane

The number of distinct degree sequences occurring in the polyhedra counted by A000109. - Sean A. Irvine, Sep 17 2017

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory, Series B, Volume 66, Issue 1, January 1996, Pages 87-122.

Cf. A000109.

a(17)-a(19) from Sean A. Irvine, Sep 17 2017

A058789 Number of polyhedra with n faces and n+1 vertices (or n vertices and n+1 faces).

Original entry on oeis.org

0, 1, 2, 11, 74, 633, 6134, 64439, 709302, 8085725, 94713809, 1134914458, 13865916560, 172301697581, 2173270387051

Offset: 4

Gerard P. Michon, Nov 30 2000

Through a(18) the only primes are 2, 11, and 64439. - Jonathan Vos Post, Apr 23 2011

			a(5)=1 because the triangular prism is the only pentahedron with 6 vertices.

G. P. Michon, Counting Polyhedra

Cf. A002856, A000109, A058786, A058787, A058788.

cp's OEIS Frontend

A007030 Non-Hamiltonian simplicial polyhedra with n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

A115340 Number of dual Hamiltonian cubic polyhedra or planar 3-connected Yutsis graphs on 2n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A253882 Number of 3-connected planar triangulations of the sphere with n vertices up to orientation preserving isomorphisms.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A133236 Number of bipartite planar graphs with 2n nodes and at least one zero eigenvector.

Views

Author

Keywords

References

Crossrefs

A133237 Number of bipartite planar graphs with 2n nodes and exactly one zero eigenvector.

Views

Author

Keywords

References

Crossrefs

A222318 Number of 4-dimensional simplicial convex polytopes with n nodes.

Views

Author

Keywords

Links

Crossrefs

Extensions

A342971 Non-1-tough simplicial polyhedra with n nodes.

Views

Author

Keywords

Comments

Crossrefs

Formula

A007020 Maximal planar degree sequences with n nodes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A058789 Number of polyhedra with n faces and n+1 vertices (or n vertices and n+1 faces).

Views

Author

Keywords

Comments