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.

A326204 Number of Hamiltonian labeled n-vertex digraphs (with loops).

Original entry on oeis.org

0, 2, 4, 120, 19104
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

			The a(2) = 4 digraph edge-sets:
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}

Links

Crossrefs

The unlabeled case is A326226.
The case without loops is A326219.
The undirected case (without loops) is A326208.
Non-Hamiltonian digraphs are A326220.
Digraphs containing a Hamiltonian path are A326214.
Cf. A000595, A002416, A003024, A003216, A057864.

Programs

  • Mathematica
    Table[Length[Select[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) = 19200 which is incorrect *)

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

A326208 Number of Hamiltonian labeled simple graphs with n vertices.

Original entry on oeis.org

0, 1, 0, 1, 10, 218, 10078, 896756, 151676112, 47754337568, 28229412456056, 31665593711174080
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

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

Links

Crossrefs

The unlabeled version is A003216.
The directed version is A326204 (with loops) or A326219 (without loops).
Simple graphs not containing a Hamiltonian cycle are A326207.
Simple graphs containing a Hamiltonian path are A326206.
Cf. A003216, A006125, A057864, A283420.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{2}]],FindHamiltonianCycle[Graph[Range[n],#]]!={}&]],{n,0,4}] (* Mathematica 8.0+ *)

Formula

A006125(n) = a(n) + A326207(n).

Extensions

a(7)-a(11) added using tinygraph by Falk Hüffner, Jun 21 2019

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

A326217 Number of labeled n-vertex digraphs (without loops) containing a Hamiltonian path.

Original entry on oeis.org

0, 0, 3, 48, 3324, 929005, 1014750550, 4305572108670
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Examples

			The a(3) = 48 edge-sets:
  {12,23}  {12,13,21}  {12,13,21,23}  {12,13,21,23,31}  {12,13,21,23,31,32}
  {12,31}  {12,13,23}  {12,13,21,31}  {12,13,21,23,32}
  {13,21}  {12,13,31}  {12,13,21,32}  {12,13,21,31,32}
  {13,32}  {12,13,32}  {12,13,23,31}  {12,13,23,31,32}
  {21,32}  {12,21,23}  {12,13,23,32}  {12,21,23,31,32}
  {23,31}  {12,21,31}  {12,13,31,32}  {13,21,23,31,32}
           {12,21,32}  {12,21,23,31}
           {12,23,31}  {12,21,23,32}
           {12,23,32}  {12,21,31,32}
           {12,31,32}  {12,23,31,32}
           {13,21,23}  {13,21,23,31}
           {13,21,31}  {13,21,23,32}
           {13,21,32}  {13,21,31,32}
           {13,23,31}  {13,23,31,32}
           {13,23,32}  {21,23,31,32}
           {13,31,32}
           {21,23,31}
           {21,23,32}
           {21,31,32}
           {23,31,32}

Links

Crossrefs

The undirected case is A326206.
The unlabeled undirected case is A057864.
The case with loops is A326214.
Unlabeled digraphs with a Hamiltonian path are A326221.
Digraphs (without loops) not containing a Hamiltonian path are A326216.
Digraphs (without loops) containing a Hamiltonian cycle are A326219.
Cf. A000595, A002416, A003024, A003216, A283420, A326204, A326208, A326213, A326225.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Tuples[Range[n],2],UnsameQ@@#&]],FindHamiltonianPath[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,4}] (* Mathematica 10.2+ *)

Formula

A053763(n) = a(n) + A326216(n).

Extensions

a(5)-a(7) from Bert Dobbelaere, Feb 21 2023

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).

A326240 Number of Hamiltonian labeled n-vertex graphs with loops.

Original entry on oeis.org

0, 2, 0, 8, 160, 6976, 644992
Offset: 0

Views

Author

Gus Wiseman, Jun 17 2019

Keywords

Comments

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

Examples

			The a(3) = 8 edge-sets:
  {12,13,23}  {11,12,13,23}  {11,12,13,22,23}  {11,12,13,22,23,33}
              {12,13,22,23}  {11,12,13,23,33}
              {12,13,23,33}  {12,13,22,23,33}

Crossrefs

The unlabeled case is A326215.
The directed case is A326204 (with loops) or A326219 (without loops).
The case without loops A326208.
Graphs with loops not containing a Hamiltonian cycle are A326239.
Cf. A000088, A003216, A006125, A057864, A283420.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Tuples[Range[n],2],OrderedQ]],FindHamiltonianCycle[Graph[Range[n],#]]!={}&]],{n,0,5}]

Formula

a(n) = A326208(n) * 2^n.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.