A conjecture

[Max Alekseyev]

Vladeta,

This sounds easy due to the following aspects:

1) "4-cycle-free graphs on n nodes with clique number 2" is equivalent
to "n-node graphs with no cycles of length less than 5"

2) According to http://mathworld.wolfram.com/EulerTransform.html
"In graph theory, if a_n is the number of unlabeled connected graphs
on n nodes satisfying some property, then b_n is the total number of
unlabeled graphs (connected or not) with the same property."

The only discrepancy left is the "biconnected" property of A126757. It
seems that if it were simply "connected" then your conjecture would
follow right away.
I'll try to figure out later what's going on here.

Regards,
Max




On Mon, Jun 16, 2008 at 5:52 PM, Vladeta Jovovic <vladeta at eunet.yu> wrote:
>
> 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,
>> %A A126757 Vladeta Jovovic (vladeta at Eunet.yu), Jun 16 2008
>
>
> Regards to all,
> Vladeta
>
>