The Four-Color Theorem of Map-Making Proved by Construction

Abstract

The objective of this paper is to prove by simple construction, generalized by induction, that the bounded areas on any map, such as found on the surface of a sheet of paper or a spherical globe, can be colored completely with just 4 distinct colors. Rather than following the tradition of examining each of tens of thousands of designs that can be produced on a planar surface, the approach here is to all the ways that any given plane, or any given part of a plane, can be divided into an old portion bearing its original color as contrasted with a new portion bearing a different color and being completely separated from the former colored portion. It is shown that for every possible manner of completely carving out any piece of any planar surface by an indexical vector, the adjacent pieces of the map, defined as ones sharing some segment of one of their borders of a length greater than 0, can always be colored with just 4 colors in a way that differentiates all the distinct pieces of the map no matter how complex or numerous the pieces may become.