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   Re: Polyhedron faces
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   Author  Topic: Re: Polyhedron faces  (Read 952 times)
Aryabhatta
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Re: Polyhedron faces  
« on: Mar 22nd, 2007, 12:42am »
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Given a polyhedron, form a Graph G as follows:
 
For each face, add a vertex to G. If faces f and g share an edge in the polyhedron, add an edge in G between corresponding vertices.
 
The degrees of a vertex is same as the number of sides to the corresponding face.
 
It is easy to show that any connected graph with 2 or more vertices has two vertices of same degree.
« Last Edit: Mar 22nd, 2007, 12:48am by Aryabhatta » IP Logged
Grimbal
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Re: Polyhedron faces  
« Reply #1 on: Mar 22nd, 2007, 1:39am »
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If there are N faces, each can have 3 to N-1 neighbours.  That is N-3 values for N items.  The pigeonhole principle tells there are at least 3 collisions, for instance 4 of the same type, 3 pairs, or one triplet and one pair.
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anant
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Re: Polyhedron faces  
« Reply #2 on: Mar 22nd, 2007, 10:34pm »
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It is easy to show that any connected graph with 2 or more vertices has two vertices of same degree.  
 
This again is from Pigeon Hole Principle
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shishir85
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Re: Polyhedron faces  
« Reply #3 on: Mar 23rd, 2007, 9:55am »
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on Mar 22nd, 2007, 10:34pm, anant wrote:
It is easy to show that any connected graph with 2 or more vertices has two vertices of same degree.  
 
This again is from Pigeon Hole Principle

 
Can you please explain this? Thanks.
 
Also, aryabhatta's solution seems to hold not only for convex polyhedra (as was originally asked) but for all kinds of polyhedra. Am I right?  
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TenaliRaman
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Re: Polyhedron faces  
« Reply #4 on: Mar 23rd, 2007, 12:32pm »
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on Mar 23rd, 2007, 9:55am, shishir85 wrote:

Can you please explain this?

n nodes, possible degrees of each node between 0 and n-1, the graph is connected, hence proved.
 
-- AI
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Aryabhatta
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Re: Polyhedron faces  
« Reply #5 on: Mar 23rd, 2007, 12:43pm »
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In fact, graph need not be connected.
 
n node, 0 to n-1 degrees. Consider vertex with n-1 degree -> connected to the rest. Contradiction, hence duplicate degree.
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anant
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Re: Polyhedron faces  
« Reply #6 on: Mar 27th, 2007, 11:42pm »
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I love this forum
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