[seqfan] Re: partition of a circle (WL 2).

[wl at particle.uni-karlsruhe.de]

Sorry, my sketch got scrambled, it seems.

In coordinates: P1= North(0,1), P2=South(0,-1), Q1=(1/2,0), Q2(0,1/2), 
Q3(-1/2,0), Q4(0,-1/2), F1=N-Q2, F2=S-Q4, F3=Q1-Q2, F4=Q2-Q3, F5=Q3-Q4, 
F6=Q4-Q1 and F7=Q1-Q3.

Wolfdieter

Quoting wl at particle.uni-karlsruhe.de:

> 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 ).
>
> Wolfdieter
>
>
>
> 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.
>> Neil
>>
>>
>> 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
>>
>> _______________________________________________
>>
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>>
>
>
>
>
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