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 1-10 of 12 results. Next

A003216 Number of Hamiltonian graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 3, 8, 48, 383, 6196, 177083, 9305118, 883156024, 152522187830, 48322518340547
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

a(1) could also be taken to be 0, but I prefer a(1) = 1. - N. J. A. Sloane, Oct 15 2006

References

  • J. P. Dolch, Names of Hamiltonian graphs, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 219.
  • R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Main diagonal of A325455 and of A325447 (for n>=3).
The labeled case is A326208.
The directed case is A326226 (with loops) or A326225 (without loops).
The case without loops is A326215.
Unlabeled simple graphs not containing a Hamiltonian cycle are A246446.
Unlabeled simple graphs containing a Hamiltonian path are A057864.
Cf. A000088, A006125, A283420.

Formula

A000088(n) = a(n) + A246446(n). - Gus Wiseman, Jun 17 2019

Extensions

Extended to n=11 by Brendan McKay, Jul 15 1996
a(12) from Sean A. Irvine, Mar 17 2015
a(13) from A246446 added by Jan Goedgebeur, Sep 07 2019

A057864 Number of simple traceable graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 5, 18, 91, 734, 10030, 248427, 11482572, 1000231510
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Number of undirected graphs on n nodes possessing a Hamiltonian path (not circuit).

Links

Crossrefs

Main diagonal of A309524.
The labeled case is A326206.
The directed case is A326221 (with loops).
Unlabeled simple graphs not containing a Hamiltonian path are A283420.
Unlabeled simple graphs containing a Hamiltonian cycle are A003216.
Cf. A000088, A006125, A246446, A283420, A326205, A326217.

Formula

A000088(n) = a(n) + A283420(n). - Gus Wiseman, Jun 17 2019

Extensions

a(8) and a(9) from Eric W. Weisstein, Jun 04 2004
a(10) from Eric W. Weisstein, May 27 2009
a(11) added using tinygraph by Falk Hüffner, Jan 19 2016

A283420 Number of simple (not necessarily connected) untraceable graphs on n nodes.

Original entry on oeis.org

0, 1, 2, 6, 16, 65, 310, 2316, 26241, 522596, 18766354
Offset: 1

Views

Author

Eric W. Weisstein, May 14 2017

Keywords

Links

Crossrefs

Cf. A000088 (number of simple graphs on n vertices).
Cf. A057864 (number of simple traceable graphs on n vertices).
Cf. A283421 (number of simple connected untraceable graphs on n vertices).
The labeled case is A326205.
The directed case is A326224 (with loops).
Unlabeled simple graphs not containing a Hamiltonian cycle are A246446.
Cf. A003216, A326206, A326217, A326221.

Formula

a(n) = A000088(n) - A057864(n).

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 *)

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

A326223 Number of non-Hamiltonian unlabeled n-vertex digraphs (with loops).

Original entry on oeis.org

1, 0, 7, 80, 2186
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(2) = 7 digraph edge-sets:
  {}
  {11}
  {12}
  {11,12}
  {11,21}
  {11,22}
  {11,12,22}

Links

Crossrefs

The labeled case is A326220.
The case without loops is A326222.
The undirected case is A246446 (without loops) or A326239 (with loops).
Hamiltonian unlabeled digraphs are A326226.
Unlabeled digraphs not containing a Hamiltonian path are A326224.
Cf. A000595, A002416, A003087, A003216, A283420, A326204, A326218, A326225.

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).

A326219 Number of labeled n-vertex Hamiltonian digraphs (without loops).

Original entry on oeis.org

0, 1, 1, 15, 1194
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) = 15 edge-sets:
  {12,23,31}  {12,13,21,32}  {12,13,21,23,31}  {12,13,21,23,31,32}
  {13,21,32}  {12,13,23,31}  {12,13,21,23,32}
              {12,21,23,31}  {12,13,21,31,32}
              {12,23,31,32}  {12,13,23,31,32}
              {13,21,23,32}  {12,21,23,31,32}
              {13,21,31,32}  {13,21,23,31,32}

Links

Crossrefs

The unlabeled case is A326225.
The undirected case is A326208 (without loops) or A326240 (with loops).
The case with loops is A326204.
Digraphs (without loops) not containing a Hamiltonian cycle are A326218.
Digraphs (without loops) containing a Hamiltonian path are A326217.
Cf. A000595, A002416, A003024, A003216, A053763, A246446, A326220, A326226.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Tuples[Range[n],2],UnsameQ@@#&]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 1200 which is incorrect *)

Formula

A053763(n) = a(n) + A326218(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 1-10 of 12 results. Next

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

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