The Four Color Theorem has been proven for quite a while now, so I'm not really breaking ground there. But last night, for some reason, it popped into my head and I started thinking about it. I feel I have devised a very simple proof, but all previous proofs (of which I am aware) have required a fair amount of computer-aided calculation. Ergo, it does not pass the smell test.
While it is not a rigorous proof, it seems relatively intuitively clear. Is there something I am missing or have I actually stumbled onto something interesting? While, I'm assuming that I have missed something, it will bug me if I don't know where my logical lapse occurred. So here is my (informal) proof:
(1) Any contiguous map of simple polygons has a corresponding map of vertices with non-intersecting paths connecting them; these paths do not need to be straight lines (and in many cases cannot be straight). Each vertex maps to one polygon and each path maps to one shared edge. Such a corresponding map can be generated by following this procedure: choose a single point in each polygon to be the corresponding vertex; from each of these vertices draw a curve to the center of each shared edge of the corresponding polygon.
The curves can be drawn without intersection for any simple polygon: the edges have order*, there is a path of finite width to each edge, and an arbitrarily large number of parallel lines can pass through any finite width.
*This is to say that one can choose any edge and travel the polygon perimeter clockwise or counterclockwise about an internal point to produce an ordered sequence of edges.
(2) Considering such a map of vertices, any three mutually connected vertices reduce to a triangle.
(3) Adding a fourth mutually connected vertex requires placing the vertex either inside or outside of the triangle. In both cases, the structure reduces to the following:
(4) Adding a fifth mutually connected vertex requires placing the vertex inside one of the triangles or outside of all of them. In the first case, the path connecting to the internal vertex must intersect one of the already existing paths. In the second case, the path connecting to the external vertex must intersect one of the already existing paths.
(5) Therefore, one cannot have five mutually connected vertices without intersection. Equivalently, there cannot be five contiguous simple polygons that mutually share edges.
(6) Therefore, one never needs more than four colors to fill in a contiguous map of simple polygons, such that no two adjacent (edge-sharing) polygons have the same color.