On Hadwiger Conjecture

TL;DR

This paper introduces a novel algorithm and matrix to analyze k-chromatic graphs, leading to a proof of Hadwiger's conjecture by characterizing such graphs and optimizing contractions.

Contribution

It presents a new contraction algorithm and transparency matrix, providing a characterization of k-chromatic graphs and settling Hadwiger's conjecture.

Findings

Algorithm reduces k-chromatic graphs to largest complete graphs

Introduction of transparency matrix with specific properties

Proof of Hadwiger's conjecture based on new characterization

Abstract

We propose an algorithm to reduce a k-chromatic graph to a complete graph of largest possible order through a well defined sequence of contractions. We introduce a new matrix called transparency matrix and state its properties. We then define correct contraction procedure to be executed to get largest possible complete graph from given connected graph. We finally give a characterization for k-chromatic graphs and use it to settle Hadwigers conjecture.