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Euler’s theorem and
    applications
           Martin B ODIN
     martin.bodin@ens-lyon.org




                                 Euler’s theorem and applications – p. 1
The theorem




              Euler’s theorem and applications – p. 2
The theorem
Theorem. 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 Theorem
Proof. Consider the plane graph G.




                                     Euler’s theorem and applications – p. 3
The Theorem
Proof. Consider the plane graph G.




We consider T , a minimal graph from G, connex.




                                             Euler’s theorem and applications – p. 3
The Theorem
Proof. 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
The Theorem
Proof. Consider the plane graph G.




Then we consider the dual graph.




                                     Euler’s theorem and applications – p. 3
The Theorem
Proof. Consider the plane graph G.




Then we consider the dual graph.
And the dual D of T .



                                     Euler’s theorem and applications – p. 3
The Theorem
Proof. 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
The Theorem
Proof. Consider the plane graph G.




Now, we have eT   + eD = e.
                  e = (v − 1) + (f − 1)

                                          Euler’s theorem and applications – p. 3
The Theorem
Proof. Consider the plane graph G.




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 of
degree at more 5.




                                      Euler’s theorem and applications – p. 4
Applications
Given a plane graph, there exists an edge of
degree at more 5.
Given a finite set of points non all in the same
line, there exists a line that contains only two of
them.




                                          Euler’s theorem and applications – p. 4
Thanks For Your
      Listenning !
Any questions ?




                        Euler’s theorem and applications – p. 5

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euler theorm

  • 1. Euler’s theorem and applications Martin B ODIN martin.bodin@ens-lyon.org Euler’s theorem and applications – p. 1
  • 2. The theorem Euler’s theorem and applications – p. 2
  • 3. The theorem Theorem. 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. The Theorem Proof. Consider the plane graph G. Euler’s theorem and applications – p. 3
  • 5. The Theorem Proof. Consider the plane graph G. We consider T , a minimal graph from G, connex. Euler’s theorem and applications – p. 3
  • 6. The Theorem Proof. 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. The Theorem Proof. Consider the plane graph G. Then we consider the dual graph. Euler’s theorem and applications – p. 3
  • 8. The Theorem Proof. 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. The Theorem Proof. 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. The Theorem Proof. Consider the plane graph G. Now, we have eT + eD = e. e = (v − 1) + (f − 1) Euler’s theorem and applications – p. 3
  • 11. The Theorem Proof. Consider the plane graph G. Now, we have eT + eD = e. v−e+f =2 Euler’s theorem and applications – p. 3
  • 12. Applications Euler’s theorem and applications – p. 4
  • 13. Applications Given a plane graph, there exists an edge of degree at more 5. Euler’s theorem and applications – p. 4
  • 14. Applications Given a plane graph, there exists an edge of degree at more 5. Given a finite set of points non all in the same line, there exists a line that contains only two of them. Euler’s theorem and applications – p. 4
  • 15. Thanks For Your Listenning ! Any questions ? Euler’s theorem and applications – p. 5