four color theorem proof?

Question

I need a proof of four colour theorem of planar graph. Here Any equivalent to the Four colour theorem for non-planar graphs? given that chromatic number is up to minimum degree plus one, but I've seen that minimum degree of planar graph up to $5$. So, how we prove that a planar graph has chromatic number $4$.

Can you please give proof of four colour theorem?

Answer 1

The only known proofs of the Four Colors Theorem are based on a computer-assisted analysis of many subcases.

You may find more details and bibliography for the original (very long) papers by Appel and Haken on Wikipedia.

It is impossible to copy here the long proof.

Answer 2

For a planar graph, you can

1. Embed the graph in the plane.

2. Draw the dual graph (again in the plane).

3. Apply the standard four-color theorem to the "map" defined by the REGIONS of the dual graph; the resulting coloring then gives the coloring for the nodes of your planar graph (which has one node per region of the dual graph).

It's hard to believe that this is what you wanted, but ...