The Four Color Theorem -- A New Simple Proof by Induction

Abstract

In 1976, Appel and Haken achieved a major break through by proving the four color theorem $(4CT)$. Their proof is based on studying a large number of cases for which a computer-assisted search for hours is required. In 1997, Robertson, Sanders, Seymour and Thomas reproved the 4CT with less need for computer verification. In this paper, we present a simple proof to the $4CT$ based on mathematical induction and contraction. We consider, in our proof, possible colorings of a minimum degree vertex, its adjacent vertices and adjacent vertices of these adjacent vertices of simple planar graphs.