A079565 Number of unlabeled and connected graphs on n vertices which are either bipartite or co-bipartite.

1, 1, 2, 6, 16, 49, 129, 481, 1845, 9506, 57896, 463909, 4769436, 65179170, 1187099045, 29082860878, 960963147303, 42920936851975, 2594399793419459, 212465886865393053, 23596018831885668391, 3557502387712889568013, 728850489548729072323085

Offset

1,3

Comments

G is bipartite iff the vertices can be partitioned into two sets such that all the edges in the graph go from one of these sets to the other. G is cobipartite iff the complement of G is bipartite.

For n >= 5, no graph can be both bipartite and co-bipartite. - Falk Hüffner, Jan 22 2016

Links

Andrew Howroyd, Table of n, a(n) for n = 1..50

Formula

For n >= 5, a(n) = A079571(n) + A005142(n). - Falk Hüffner, Jan 22 2016

Example

Let G be a graph with 5 vertices, 4 of which form a path and the 5th adjacent only to the two vertices in the middle of the path. Then G is not bipartite nor cobipartite because there is a triangle in both G and its complement.

Mathematica

A005142 = Import["https://oeis.org/A005142/b005142.txt", "Table"][[All, 2]];
A033995 = Import["https://oeis.org/A033995/b033995.txt", "Table"][[All, 2]];
a[n_] := If[n<5, {1, 1, 2, 6}[[n]], A005142[[n+1]] + A033995[[n+1]] - Floor[n/2]];
a /@ Range[1, 50] (* Jean-François Alcover, Sep 17 2019 *)

Crossrefs

Cf. A005142, A079571.

Sequence in context: A272411 A151528 A132803 * A052890 A052814 A192401

Adjacent sequences: A079562 A079563 A079564 * A079566 A079567 A079568

Keyword

nonn

Author

Jim Nastos, Jan 24 2003

Extensions

More terms using formula by Falk Hüffner, Jan 22 2016

Terms a(21) and beyond from Andrew Howroyd, Sep 05 2018

Status

approved