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

## TL;DR

This paper introduces a new simple proof of the four color theorem using induction and contraction, simplifying previous complex proofs that relied heavily on computer verification.

## Contribution

It offers a novel, straightforward proof of the four color theorem based on induction and contraction methods, reducing reliance on computer-assisted case analysis.

## Key findings

- Proof based on induction and contraction techniques.
- Simplifies the understanding of the four color theorem.
- Reduces computational verification needed in previous proofs.

## 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.