To Prove Four Color Theorem

TL;DR

This paper presents a non-computer-assisted proof of the four color theorem, extends the approach to Hadwiger's conjecture, and introduces algorithms for graph coloring and planarity testing.

Contribution

It offers a novel, generalizable proof of the four color theorem and Hadwiger Conjecture, along with algorithms applicable to broader classes of graphs.

Findings

Proof of four color theorem without computer assistance

Algorithms for coloring and planarity testing of planar graphs

Extension of methods to graphs with $K_x$ minors

Abstract

In this paper, we give a proof for four color theorem(four color conjecture). Our proof does not involve computer assistance and the most important is that it can be generalized to prove Hadwiger Conjecture. Moreover, we give algorithms to color and test planarity of planar graphs, which can be generalized to graphs containing $K_x(x>5)$ minor. There are four parts of this paper: Part-1: To Prove Four Color Theorem Part-2: An Equivalent Statement of Hadwiger Conjecture when $k=5$ Part-3: A New Proof of Wagner's Equivalence Theorem Part-4: A Geometric View of Outerplanar Graph