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