Blueprint for a Classic Proof of the Four Colour Theorem

TL;DR

This paper presents a novel proof of the Four Colour Theorem based on triangle orientations and their sums around vertices in triangulated planar graphs, offering a new perspective on classic graph coloring.

Contribution

It introduces a unique proof method using triangle orientations and their sums, providing a fresh approach to the Four Colour Theorem.

Findings

The proof demonstrates that triangle orientations can ensure four-colorability of planar graphs.

It establishes that the sum of orientations around vertices is a multiple of 3, leading to a valid coloring.

The method reconstructs the graph while maintaining orientation conditions, ensuring the theorem holds.

Abstract

The proof uses the property that the vertices of a triangulated planar graph can be four coloured if the triangles can have a +1 or -1 orientation so that the sum of the triangle orientations around each vertex is a multiple of 3. Such orientation is first used separately on one of the two triangulated polygons resulting from a Hamilton circuit in a triangulated planar graph with v vertices. The graph is then reconstructed by adding the triangles of the other polygon one by one. When the graph is totally reconstructed there is always a combination for the orientations of the triangles for which their sum around each of v-2 successive vertices in the Hamilton circuit is a multiple of 3. It is then provable that the sum of the triangle orientations around the two remaining vertices must also be a multiple of 3.