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

[William Orrick]

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
> >
> > --
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> >
>
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