Euler’s theorem and    applications           Martin B ODIN     martin.bodin@ens-lyon.org                                 ...
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,  ...
The TheoremProof. Consider the plane graph G.                                     Euler’s theorem and applications – p. 3
The TheoremProof. Consider the plane graph G.We consider T , a minimal graph from G, connex.                              ...
The TheoremProof. Consider the plane graph G.We consider T , a minimal graph from G, connex.T is a tree.Thus eT = v − 1, w...
The TheoremProof. Consider the plane graph G.Then we consider the dual graph.                                     Euler’s ...
The TheoremProof. Consider the plane graph G.Then we consider the dual graph.And the dual D of T .                        ...
The TheoremProof. Consider the plane graph G.Then we consider the dual graph.And the dual D of T .D in a also a tree.Thus ...
The TheoremProof. Consider the plane graph G.Now, we have eT   + eD = e.                  e = (v − 1) + (f − 1)           ...
The TheoremProof. Consider the plane graph G.Now, we have eT   + eD = e.                     v−e+f =2                     ...
Applications               Euler’s theorem and applications – p. 4
ApplicationsGiven a plane graph, there exists an edge ofdegree at more 5.                                      Euler’s the...
ApplicationsGiven a plane graph, there exists an edge ofdegree at more 5.Given a finite set of points non all in the sameli...
Thanks For Your      Listenning !Any questions ?                        Euler’s theorem and applications – p. 5
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euler theorm

  1. 1. Euler’s theorem and applications Martin B ODIN martin.bodin@ens-lyon.org Euler’s theorem and applications – p. 1
  2. 2. The theorem Euler’s theorem and applications – p. 2
  3. 3. 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
  4. 4. The TheoremProof. Consider the plane graph G. Euler’s theorem and applications – p. 3
  5. 5. The TheoremProof. Consider the plane graph G.We consider T , a minimal graph from G, connex. Euler’s theorem and applications – p. 3
  6. 6. The TheoremProof. Consider the plane graph G.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
  7. 7. The TheoremProof. Consider the plane graph G.Then we consider the dual graph. Euler’s theorem and applications – p. 3
  8. 8. The TheoremProof. Consider the plane graph G.Then we consider the dual graph.And the dual D of T . Euler’s theorem and applications – p. 3
  9. 9. The TheoremProof. Consider the plane graph G.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
  10. 10. The TheoremProof. Consider the plane graph G.Now, we have eT + eD = e. e = (v − 1) + (f − 1) Euler’s theorem and applications – p. 3
  11. 11. The TheoremProof. Consider the plane graph G.Now, we have eT + eD = e. v−e+f =2 Euler’s theorem and applications – p. 3
  12. 12. Applications Euler’s theorem and applications – p. 4
  13. 13. ApplicationsGiven a plane graph, there exists an edge ofdegree at more 5. Euler’s theorem and applications – p. 4
  14. 14. ApplicationsGiven 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
  15. 15. Thanks For Your Listenning !Any questions ? Euler’s theorem and applications – p. 5
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