[seqfan] Re: simplicial polyhedra, aka maximal planar graphs: is A000109 always equal to A007030 plus A115340?

[Allan Wechsler]

You must be right, Will, I got my arithmetic on the indices backwards.
Anyway, glad this is getting straightened out.

On Mon, Feb 22, 2021 at 11:04 PM William Orrick <will.orrick at gmail.com>
wrote:

> Dear Allan,
>
> Thanks for your comments. Some changes have already been made to two of the
> sequences based on advice from Brendan McKay, and the issue seems to have
> been settled. Sorry for not updating the list earlier--I somehow missed
> that this post had appeared.
>
> Doesn't the minimal non-Hamiltonian example have 11 vertices, not 13? I'm
> thinking of the Goldner-Harary graph. Wikipedia has an image, but not the
> best representation in my opinion since it doesn't make the 3-fold symmetry
> obvious. A representation as a deltahedron could be obtained by gluing a
> tetrahedron to each of the six faces of a triangular bipyramid.
>
> Best,
> Will
>
> On Sun, Feb 21, 2021 at 2:36 AM Allan Wechsler <acwacw at gmail.com> wrote:
>
> > Just going by the text at all three sequences, the identity looks right,
> > although I think different indexing choices were made for  the three
> > A115340. William P. Orrick has already contributed the identity at
> A115340.
> > A000109 counts all the fully-triangulated polyhedra with n vertices;
> > A115340 counts the number with n+2 vertices that admit a Hamiltonian
> cycle
> > (which is most of them, for small n); and A007030 is the rest of them,
> the
> > triangulated polyhedra on n+2 vertices that *don't *have a Hamiltonian
> > cycle. If the identity lets us fill in some values on any of the
> sequences,
> > we should do so.
> >
> > All three of these sequences would be well-served by having some examples
> > -- imagery would be especially nice. I want to see that minimal
> > non-Hamiltonian deltahedron with 13 vertices.
> >
> > On Sat, Feb 20, 2021 at 5:40 AM William Orrick <will.orrick at gmail.com>
> > wrote:
> >
> > > Dear SeqFans,
> > >
> > > The subject line says it all. I suspect, but am not sufficiently
> > conversant
> > > with definitions to be sure, that the relation should always hold. If
> > > true, A007030 can be extended. Some cross refs could also be added to
> all
> > > three sequences.
> > >
> > > -Will
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
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> >
>
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