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.

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).

A326206 Number of n-vertex labeled simple graphs containing a Hamiltonian path.

Original entry on oeis.org

0, 0, 1, 4, 34, 633, 23368, 1699012, 237934760, 64558137140, 34126032806936, 35513501049012952
Offset: 0

Views

Author

Gus Wiseman, Jun 14 2019

Keywords

Comments

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

Links

Crossrefs

The unlabeled case is A057864.
The directed case is A326214 (with loops) or A326217 (without loops).
Simple graphs without a Hamiltonian path are A326205.
Simple graphs with a Hamiltonian cycle are A326208.
Cf. A003216, A006125, A057864, A283420.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{2}]],FindHamiltonianPath[Graph[Range[n],#]]!={}&]],{n,0,4}] (* Mathematica 10.2+ *)

Formula

A006125(n) = a(n) + A326205(n).

Extensions

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

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).

A326214 Number of labeled n-vertex digraphs (with loops) containing a (directed) Hamiltonian path.

Original entry on oeis.org

0, 0, 12, 384, 53184
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

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

Examples

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

Links

Crossrefs

The unlabeled case is A326221.
The undirected case is A326206.
The case without loops is A326217.
Digraphs not containing a Hamiltonian path are A326213.
Digraphs containing a Hamiltonian cycle are A326204.
Cf. A000595, A002416, A003024, A003216, A057864, A326224, A326225, A326226.

Programs

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

Formula

A002416(n) = a(n) + A326213(n).

A326216 Number of labeled n-vertex digraphs (without loops) not containing a (directed) Hamiltonian path.

Original entry on oeis.org

1, 1, 1, 16, 772
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

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

Examples

			The a(3) = 16 edge-sets:
  {}  {12}  {12,13}
      {13}  {12,21}
      {21}  {12,32}
      {23}  {13,23}
      {31}  {13,31}
      {32}  {21,23}
            {21,31}
            {23,32}
            {31,32}

Links

Crossrefs

Unlabeled digraphs not containing a Hamiltonian path are A326224.
The undirected case is A326205.
The unlabeled undirected case is A283420.
The case with loops is A326213.
Digraphs (without loops) containing a Hamiltonian path are A326217.
Digraphs (without loops) not containing a Hamiltonian cycle are A326218.
Cf. A000595, A002416, A003024, A003216, A057864, A326206, A326214, A326220, A326221, 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) + A326217(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.

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

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