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Same Degree Vertices (Posted on 2008-08-21) Difficulty: 3 of 5
Define a finite graph (V,E) where V is a finite, non-empty set and
E is a subset of { {a,b} | a,b in V and a ≠ b }.

If c in V, we define

   E(c) = { {a,c} in E | a in V }

   V(c) = { a in V | {a,c} in E }

   d(c) = |E(c)| = |V(c)|
Prove for any finite graph (V,E), with |V| > 1, that
there exists a,b in V such that a ≠ b and d(a) = d(b).

   V is the set of vertices of the graph.

   E is the set of edges of the graph.

   E(c) is the set of edges incident to vertex c.

   V(c) is the set of vertices adjacent to vertex c.

   d(c) is the degree of vertex c.

See The Solution Submitted by Bractals    
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  Subject Author Date
SolutionsolutionPaul2008-08-22 22:07:58
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