cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A246446 Number of nonhamiltonian graphs with n nodes.

Original entry on oeis.org

0, 2, 3, 8, 26, 108, 661, 6150, 97585, 2700050, 135841840, 12568984762, 2179513027405
Offset: 1

Views

Author

Eric W. Weisstein, Aug 26 2014

Keywords

Links

Crossrefs

Cf. A000088 (number of simple graphs on n nodes).
Cf. A003216 (number of Hamiltonian graphs on n nodes).
Cf. A126149 (number of connected nonhamiltonian graphs on n nodes).
The labeled case is A326207.
The directed case is A326223 (with loops) or A326222 (without loops).
Unlabeled simple graphs not containing a Hamiltonian path are A283420.
Cf. A006125, A057864, A326205, A326206.

Programs

Formula

a(n) = A000088(n) - A003216(n).

Extensions

a(12) from formula by Falk Hüffner, Aug 13 2017
a(13) added by Jan Goedgebeur, May 07 2019

A326225 Number of Hamiltonian unlabeled n-vertex digraphs (without loops).

Original entry on oeis.org

0, 1, 1, 4, 61, 3725, 844141, 626078904
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			Non-isomorphic representatives of the a(3) = 4 digraph edge-sets:
  {12,23,31}
  {12,13,21,32}
  {12,13,21,23,31}
  {12,13,21,23,31,32}

Links

Crossrefs

The labeled case is A326219.
The case with loops is A326226.
The undirected case is A003216.
Non-Hamiltonian unlabeled digraphs (without loops) are A326222.
Cf. A000595, A002416, A003087, A003216, A053763, A246446, A326204, A326218, A326223.

Extensions

a(5)-a(7) from Sean A. Irvine, Jun 16 2019

A326226 Number of unlabeled n-vertex Hamiltonian digraphs (with loops).

Original entry on oeis.org

0, 2, 3, 24, 858
Offset: 0

Views

Author

Gus Wiseman, Jun 14 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			Non-isomorphic representatives of the a(2) = 3 digraph edge-sets:
  {12,21}
  {11,12,21}
  {11,12,21,22}

Links

Crossrefs

The labeled case is A326204.
The case without loops is A326225.
The undirected case is A003216 (without loops) or A326215 (with loops).
Unlabeled non-Hamiltonian digraphs are A326223.
Unlabeled digraphs with a Hamiltonian path are A326221.
Cf. A000595, A002416, A003087, A057864, A246446, A326208.

Programs

  • Mathematica
    dinorm[m_]:=If[m=={},{},If[Union@@m!=Range[Max@@Flatten[m]],dinorm[m/. Apply[Rule,Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}],{1}]],First[Sort[dinorm[m,1]]]]];
dinorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#1>=aft&]}]},Union@@(dinorm[#1,aft+1]&)/@Union[Table[Map[Sort,m/. {par+aft-1->aft,aft->par+aft-1},{0}],{par,First/@Position[mx,Max[mx]]}]]]];
Table[Length[Select[Union[dinorm/@Subsets[Tuples[Range[n],2]]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 867 which is incorrect *)

A326224 Number of unlabeled n-vertex digraphs (with loops) not containing a Hamiltonian path.

Original entry on oeis.org

1, 2, 3, 30, 649
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2019

Keywords

Comments

A directed path is Hamiltonian if it passes through every vertex exactly once.

Links

Crossrefs

The labeled case is A326213.
The undirected case is A283420 (without loops).
Unlabeled digraphs containing a Hamiltonian path are A326221.
Unlabeled digraphs not containing a Hamiltonian cycle are A326223.
Cf. A000595, A002416, A003087, A003216, A057864.

Formula

A000595(n) = a(n) + A326221(n).

A326220 Number of non-Hamiltonian labeled n-vertex digraphs (with loops).

Original entry on oeis.org

1, 0, 12, 392, 46432, 20023232, 30595305216
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			The a(2) = 12 digraph edge-sets:
  {}  {11}  {11,12}  {11,12,22}
      {12}  {11,21}  {11,21,22}
      {21}  {11,22}
      {22}  {12,22}
            {21,22}

Links

Crossrefs

The unlabeled case is A326223.
The undirected case is A326239 (with loops) or A326207 (without loops).
The case without loops is A326218.
Digraphs (with loops) containing a Hamiltonian cycle are A326204.
Digraphs (with loops) not containing a Hamiltonian path are A326213.
Cf. A000595, A002416, A003024, A003216, A246446, A326208, A326219, A326222, A326224.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Tuples[Range[n],2]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]=={}&]],{n,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 46336 which is incorrect *)

Extensions

a(5)-a(6) from Bert Dobbelaere, Jun 11 2024

A326218 Number of non-Hamiltonian labeled n-vertex digraphs (without loops).

Original entry on oeis.org

1, 0, 3, 49, 2902
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			The a(3) = 49 edge-sets:
  {}  {12}  {12,13}  {12,13,21}  {12,13,21,23}
      {13}  {12,21}  {12,13,23}  {12,13,21,31}
      {21}  {12,23}  {12,13,31}  {12,13,23,32}
      {23}  {12,31}  {12,13,32}  {12,13,31,32}
      {31}  {12,32}  {12,21,23}  {12,21,23,32}
      {32}  {13,21}  {12,21,31}  {12,21,31,32}
            {13,23}  {12,21,32}  {13,21,23,31}
            {13,31}  {12,23,32}  {13,23,31,32}
            {13,32}  {12,31,32}  {21,23,31,32}
            {21,23}  {13,21,23}
            {21,31}  {13,21,31}
            {21,32}  {13,23,31}
            {23,31}  {13,23,32}
            {23,32}  {13,31,32}
            {31,32}  {21,23,31}
                     {21,23,32}
                     {21,31,32}
                     {23,31,32}

Links

Crossrefs

The unlabeled case is A326222.
The undirected case is A326207.
The case with loops is A326220.
Digraphs (without loops) containing a Hamiltonian cycle are A326219.
Digraphs (without loops) not containing a Hamiltonian path are A326216.
Cf. A000595, A002416, A003024, A003216, A246446, A326204, A326213, A326223, A326225.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Tuples[Range[n],2],UnsameQ@@#&]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]=={}&]],{n,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 2896 which is incorrect *)

Formula

A053763(n) = a(n) + A326219(n).

A326222 Number of non-Hamiltonian unlabeled n-vertex digraphs (without loops).

Original entry on oeis.org

1, 0, 2, 12, 157, 5883, 696803, 255954536
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Links

Crossrefs

The labeled case is A326218 (without loops) or A326220 (with loops).
The undirected case (without loops) is A246446.
The case with loops is A326223.
Hamiltonian unlabeled digraphs are A326225 (without loops) or A003216 (with loops).
Cf. A000273, A000595, A002416, A003087, A053763, A326216, A326217, A326224, A326226.

Formula

a(n) = A000273(n) - A326225(n). - Pontus von Brömssen, Mar 17 2024

Extensions

a(5)-a(7) (using A000273 and A326225) from Pontus von Brömssen, Mar 17 2024
Showing 1-7 of 7 results.

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