[seqfan] Re: partition of a circle
njasloane at gmail.com
Sat May 12 00:56:01 CEST 2012
Wolfdieter, Yes, I agree that we need more conditions
that will rule out your example.
In fact in the Valette-Z. paper, they add the condition
that the partition must be admissible, which
they define to mean that if K and L are two distinct pieces,
then the intersection (boundary of K) intersect (boundary of L)
This rules out your example, since the two big pieces has intersection
which is not connected.
By the way, they do NOT require that the pieces be convex. (That would be
another way to
rule out your example)
On Fri, May 11, 2012 at 11:35 AM, <wl at particle.uni-karlsruhe.de> wrote:
> Neil, but then what about n=4 and the graph with b=2, i=4 internal nodes
> Q1 to Q4 with e=7 internal edges F1 to F7. See below. This is not among
> the 15 paper drawings. If Q1 and Q3 would be connected to some two new
> circle nodes should it be admitted for n=6?
> Cycles for inner nodes are allowed (see n=4 paper examples).
> A sketch without the circle line:
> P1 | Q2
> / \ Q3---Q1 \ / Q4
> | P2
> Besides the Euler rule you gave earlier (n = e-i+1, here 4 = 7 - 4 + 1)
> one can also use the degree check formula: 2*|edges| = sum over the degrees
> of all nodes, that is
> 2(b+e) =sum(dP_j,j=1..b) + sum(dQ_j,j=1..i).
> Here: 2*(2 + 7) = 18 = 2*3 + 4*3 = 6 + 12.
> (In your grid example (actually a 12-gon with 16 part, where the four
> square nodes of degree 2 have been deleted) it was 2*(12 + 24) = 72 = 12*3
> + 9*4 ).
> Quoting Neil Sloane <njasloane at gmail.com>:
> Wolfdieter, I don't think that rule is needed.
>> I think the following is a legitimate dissection (into 16 regions):
>> draw a square grid with 5 horizontal lines and 5 vertical lines.
>> The boundary is a square, but topologically that is the same as
>> a circle. There are 12 P's and 9 Q's.
>> I asked Tudor Z. if he knew the next term, but he said no.
>> On Fri, May 11, 2012 at 3:10 AM, <wl at particle.uni-karlsruhe.de> wrote:
>> Maybe one should also include the rule:
>>> There has to be at least one line segment (edge) connecting each internal
>>> node Q to a boundary node P. This would exclude Andrews n=3 and n=5
>>> graphs, and also Andrew's n=5 graph with all of the four 'outer' Qs
>>> connected to some Ps (now n=8), but the 'central' node Q is not connected
>>> to any P. Such a central Q is never connectable to any P (planarity).
>>> I have written yesterday to one of the authors (T. Zamfirescu at Dortmund
>>> TH) my zeroth order guess of the rules, and asked him for a definition of
>>> 'Jordan domain' on a euclidean plane E_2 (maybe some fans can here help
>>> also). I am waiting for his answer.
>>> Wolfdieter Lang
>>> Seqfan Mailing list - http://list.seqfan.eu/
>> Dear Friends, I will soon be retiring from AT&T. New coordinates:
>> Neil J. A. Sloane, President, OEIS Foundation
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>> Seqfan Mailing list - http://list.seqfan.eu/
> Seqfan Mailing list - http://list.seqfan.eu/
Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com
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