# A conjecture

Brendan McKay bdm at cs.anu.edu.au
Tue Jun 17 04:17:30 CEST 2008

```Hi, I don't understand the definition.  By "clique number 2" do you
mean that the largest clique has 2 vertices?  Then clique number 2
plus 4-cycle-free means that the girth is at least 5, right?
However the numbers of biconnected graphs with girth at least 5,
starting with 5 vertices, are
1, 1, 2, 5, 12, 40, 154, 766, 4672, 34547, 302185, 3073267, 35933851
which is not in OEIS.

Probably I am missing something in the definition.  In case it helps,
here are the ones I have of order 8:

0: 4 6; 1: 5 6; 2: 5 7; 3: 6 7; 4: 0 7;   5: 1 2;   6: 0 1 3; 7: 2 3 4;
0: 4 5; 1: 4 6; 2: 5 7; 3: 6 7; 4: 0 1;   5: 0 2;   6: 1 3;   7: 2 3;
0: 4 5; 1: 4 6; 2: 5 7; 3: 6 7; 4: 0 1 7; 5: 0 2;   6: 1 3;   7: 2 3 4;
0: 4 5; 1: 4 7; 2: 5 6; 3: 6 7; 4: 0 1 6; 5: 0 2 7; 6: 2 3 4; 7: 1 3 5;
0: 3 6; 1: 4 6; 2: 5 6; 3: 0 7; 4: 1 7;   5: 2 7;   6: 0 1 2; 7: 3 4 5;

Cheers, Brendan.

>
> Seqfans,
>
> >Subject: COMMENT FROM Vladeta Jovovic RE A126757
> >
> >
> >%I A126757
> >%S A126757 0, 1, 1, 2, 4, 8, 18, 47, 137, 464, 1793, 8167, 43645, 275480
> >%N A126757 Number of biconnected 4-cycle-free graphs on n nodes with
> >clique number 2.
> >%C A126757 Conjecture: Inverse Euler transform of A006787.
> >%O A126757 1
> >%K A126757 ,nonn,