euler theorm
TRANSCRIPT
Euler’s theorem andapplications
Martin BODIN
Euler’s theorem and applications – p. 1
The theorem
Euler’s theorem and applications – p. 2
The theoremTheorem. Given a plane graph, if v is the number of vertex,e, the number of edges, and f the number of faces,
v − e + f = 2
Euler’s theorem and applications – p. 2
The TheoremProof. Consider the plane graph G.b
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Euler’s theorem and applications – p. 3
The TheoremProof. Consider the plane graph G.b
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We consider T , a minimal graph from G, connex.
Euler’s theorem and applications – p. 3
The TheoremProof. Consider the plane graph G.b
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We consider T , a minimal graph from G, connex.T is a tree.Thus eT = v − 1, where eT is the number of T ’s edge.
Euler’s theorem and applications – p. 3
The TheoremProof. Consider the plane graph G.b
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Then we consider the dual graph.
Euler’s theorem and applications – p. 3
The TheoremProof. Consider the plane graph G.b
b
b
b
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Then we consider the dual graph.And the dual D of T .
Euler’s theorem and applications – p. 3
The TheoremProof. Consider the plane graph G.b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
Then we consider the dual graph.And the dual D of T .D in a also a tree.Thus eD = f − 1.
Euler’s theorem and applications – p. 3
The TheoremProof. Consider the plane graph G.b
b
b
b
b
b
b
b
b
b
b
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Now, we have eT + eD = e.
e = (v − 1) + (f − 1)
Euler’s theorem and applications – p. 3
The TheoremProof. Consider the plane graph G.b
b
b
b
b
b
b
b
b
b
b
b
b
b
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b
Now, we have eT + eD = e.
v − e + f = 2
Euler’s theorem and applications – p. 3
Applications
Euler’s theorem and applications – p. 4
Applications
Given a plane graph, there exists an edge ofdegree at more 5.
Euler’s theorem and applications – p. 4
Applications
Given a plane graph, there exists an edge ofdegree at more 5.Given a finite set of points non all in the sameline, there exists a line that contains only two ofthem.
Euler’s theorem and applications – p. 4
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Euler’s theorem and applications – p. 5