- #1

Adel Makram

- 635

- 15

- TL;DR Summary
- Visual proof of the 4-colour theorem.

The 4-colour theorem states that the maximum number of colours required to paint a map is 4.

The proof requires exhaustive computation with a help of a computer.

But I thought that one can visually prove the theorem in the following way;

If one replaces the map with a graph where each region with a different colour is represented by a vertex and the two adjacent regions represented by an edge between the vertices. Then, the maximum number of vertices in any complete planar local graph where every two vertices are connected and no crossing of the edges is allowed must be 4.

For example, the picture on the right shows that all vertices (a,b,c,d) are connected and therefore corresponds to the fact that the maximum number of allowed colours is 4.

While the picture on the left shows that it is impossible to get 5 colours to paint the map because the regions represented by the vertices (a) and (d) in the map will not be adjacent to each other (no edge connects (a) and (d) exits without crossing the other edges).

The proof requires exhaustive computation with a help of a computer.

But I thought that one can visually prove the theorem in the following way;

If one replaces the map with a graph where each region with a different colour is represented by a vertex and the two adjacent regions represented by an edge between the vertices. Then, the maximum number of vertices in any complete planar local graph where every two vertices are connected and no crossing of the edges is allowed must be 4.

For example, the picture on the right shows that all vertices (a,b,c,d) are connected and therefore corresponds to the fact that the maximum number of allowed colours is 4.

While the picture on the left shows that it is impossible to get 5 colours to paint the map because the regions represented by the vertices (a) and (d) in the map will not be adjacent to each other (no edge connects (a) and (d) exits without crossing the other edges).