A New Proof of the Four Color Theorem

Abstract

The Four Color Theorem states that every finite map on the sphere can be colored with at most four colors so that any two adjacent regions receive different colors. In this paper, we study the theorem from a topological and graph-theoretic point of view and introduce a strengthened boundary coloring property for disk triangulations. By applying the inductive hypothesis, this property is preserved. This provides an inductive framework for proving the Four Color Theorem without using a computer and exhaustion.