Hadwiger conjecture for 8-coloring graph

TL;DR

This paper introduces a novel approach using chromatic space and Euler space to address the long-standing Hadwiger Conjecture, providing a feasible proof for graphs with up to 8 colors.

Contribution

It proposes a new chromatic graph configuration and a method to prove the Hadwiger Conjecture for 8-colorable graphs, advancing understanding in graph coloring theory.

Findings

Defined a chromatic plane and chromatic coordinates in Euler space.

Developed a method to prove Hadwiger Conjecture for 8-color graphs.

Presented a new framework for describing graph coloring issues.

Abstract

Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable graphs have been completely proved to convince all1-5, but the proofs are tremendously difficult for over the 5-colorable graph6,7. Although the development of graph theory inspires scientists to understand graph coloring deeply, it is still an open problem for over 7-colorable graphs6,7. Therefore, we put forward a brand new chromatic graph configuration and show how to describe the graph coloring issues in chromatic space. Based on this idea, we define a chromatic plane and configure the chromatic coordinates in Euler space. Also, we find a method to prove Hadwiger Conjecture for every 8-coloring graph feasible.