Geodesic
Blueprint for a classic proof of the 4 colour theorem
The proof uses the property that the vertices of a triangulated planar graph with v vertices can be four coloured if the triangles of it can be given a +1 or -1 orientation in such a way that the sum of the triangle orientations around each vertex is a multiple of 3 (or their sumMod3 is 0). The proof is by association of each of v-2 vertices with two triangles. Together they form trios in such a way that each triangle belongs to a trio and only to one. The trios are formed in such a way that the two remaining vertices are linked by an edge. From this association it follows that there is always a combination for the orientations of the triangles so that their sum around the v-2 vertices is a multiple of 3. In that case it is provable that the sum of the triangle orientations around the two remaining vertices must also be a multiple of 3.
Classification : 20H15 
@article{VM_2010_12_2_a2,
     author = {Patrick Labarque},
     title = {Blueprint for a classic proof of the 4 colour theorem},
     journal = {Visual Mathematics},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VM_2010_12_2_a2/}
}
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Patrick Labarque. Blueprint for a classic proof of the 4 colour theorem. Visual Mathematics, Tome 12 (2010) no. 2. http://geodesic.mathdoc.fr/item/VM_2010_12_2_a2/