Euler's theorem states that for any plane graph, the number of vertices (v) minus the number of edges (e) plus the number of faces (f) equals 2. The document proves this theorem by considering a minimal tree (T) within the graph and its dual tree (D), showing that the number of edges of T and D sum to the total edges (e) of the original graph. Some applications of the theorem are that any plane graph contains an edge of degree 5 or higher and any finite set of points not all on a line contains a line with exactly two points.

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