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 1-3 of 3 results.

A243252 Number of simple connected graphs with n nodes whose fractional chromatic number is equal to its (integer) chromatic number.

Original entry on oeis.org

1, 1, 2, 6, 20, 109, 820, 10621, 244616, 10747278
Offset: 1

Views

Author

Travis Hoppe and Anna Petrone, Jun 20 2014

Keywords

Comments

This implies that there is no difference 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. A243251 (fractional chromatic number is not equal to chromatic number).
Cf. A287007 (not necessarily connected simple graphs with fractional chromatic number equal to chromatic number).
Cf. A287008 (disconnected simple graphs with fractional chromatic number equal to chromatic number).

Formula

a(n) = A287008(n) - A287007(n).

Extensions

Name edited by Michel Marcus, Jan 12 2025

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).

A287023 Number of simple disconnected weakly perfect graphs on n vertices.

Original entry on oeis.org

0, 1, 2, 5, 13, 43, 186, 1187, 13006, 270900, 11286208
Offset: 1

Views

Author

Eric W. Weisstein, May 18 2017

Keywords

Comments

First differs from A287008 (number of simple disconnected graphs with equal chromatic and fractional chromatic number) at a(8).

Links

Crossrefs

Cf. A198634 (number of not necessarily connected simple weakly perfect graphs).
Cf. A287009 (number of not connected simple weakly perfect graphs).
Cf. A287008 (number of simple disconnected graphs with equal chromatic and fractional chromatic number).

Formula

a(n) = A198634(n) - A287009(n).

Extensions

a(11) from formula by Falk Hüffner, Aug 13 2017
Showing 1-3 of 3 results.

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