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 1-4 of 4 results.

A287007 Number of simple (not necessarily connected) graphs on n vertices whose fractional chromatic number equals its (integer) chromatic number.

Original entry on oeis.org

1, 2, 4, 11, 33, 152, 1006, 11808, 257625, 11018264
Offset: 1

Views

Author

Eric W. Weisstein, May 17 2017

Keywords

Comments

First differs from A198634 (weakly perfect graphs) at a(8). The three 8-node graphs that have equal chromatic and fractional chromatic numbers but are not weakly perfect are the 4-antiprism graph and 50- and 84-Johnson solid skeleton graphs, all of which have clique number 3 but chromatic and fractional chromatic number 4.

Links

Crossrefs

Cf. A198634 (number of weakly perfect graphs on n nodes).
Cf. A243252 (number of simple connected graphs on n nodes with fractional chromatic number equal to chromatic number).
Cf. A287008 (number of simple disconnected graphs on n nodes with fractional chromatic number equal to chromatic number).

Formula

a(n) = A243252(n) + A287008(n).

A287008 Number of disconnected simple graphs on n vertices whose fractional chromatic number equals its chromatic number.

Original entry on oeis.org

0, 1, 2, 5, 13, 43, 186, 1187, 13009, 270986
Offset: 1

Views

Author

Eric W. Weisstein, May 17 2017

Keywords

Links

Crossrefs

Cf. A287007 (number of not-necessarily connected simple graphs on n nodes with fractional chromatic number equal to chromatic number).
Cf. A243252 (number of simple connected graphs on n node with fractional chromatic number equal to chromatic number).

Formula

a(n) = A287007(n) - A243252(n).

A243251 Number of simple connected graphs with n nodes whose fractional chromatic number is not equal to its (integer) chromatic number.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 33, 496, 16464, 969293
Offset: 1

Views

Author

Travis Hoppe and Anna Petrone, Jun 20 2014

Keywords

Comments

This implies that there is a gap between the corresponding integer and linear programs defining fractional colorings. Every simple graph has a fractional chromatic number which is a rational number or integer.

Links

Crossrefs

Cf. A243252 (fractional chromatic number is equal to chromatic number)

Formula

a(n) = A001349(n) - A243252(n). - Andrew Howroyd, Nov 03 2017

Extensions

Name edited by Michel Marcus, Jan 12 2025

A287009 Number of connected simple weakly perfect graphs on n vertices.

Original entry on oeis.org

1, 1, 2, 6, 20, 109, 820, 10618, 244536, 10740858, 905808814
Offset: 1

Views

Author

Eric W. Weisstein, May 17 2017

Keywords

Comments

First differs from A243252 (connected simple graphs whose fractional number equals its chromatic number) at a(8). The three (connected) 8-node graphs that have equal chromatic and fractional chromatic numbers but are not weakly perfect are the 4-antiprism graph and 50- and 84-Johnson solid skeleton graphs, all of which have clique number 3 but chromatic and fractional chromatic number 4.

Links

  • F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 4361e42
  • Eric Weisstein's World of Mathematics, Connected Graph
  • Eric Weisstein's World of Mathematics, Weakly Perfect Graph

Crossrefs

Cf. A198634 (not necessarily connected weakly perfect simple graphs on n nodes).
Cf. A287023 (disconnected weakly perfect simple graphs on n nodes).
Cf. A243252 (connected simple graphs whose fractional number equals its chromatic number).

Formula

a(n) = A198634(n) - A287023(n).

Extensions

a(9)-a(10) from Eric W. Weisstein, May 18 2017
a(11) added using tinygraph by Falk Hüffner, Aug 13 2017
Showing 1-4 of 4 results.

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

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