A Note on Hadwiger's Conjecture

TL;DR

This paper improves the upper bound on the chromatic number of $K_{t+1}$-minor-free graphs based on their minimum degree, contributing to the understanding of Hadwiger's Conjecture in graph theory.

Contribution

It provides a slightly better upper bound on the chromatic number for $K_{t+1}$-minor-free graphs based on minimum degree, advancing the study of Hadwiger's Conjecture.

Findings

Established a new upper bound on the chromatic number.

Improved understanding of the relationship between minors and graph coloring.

Contributed to the ongoing research on Hadwiger's Conjecture.

Abstract

Hadwiger's Conjecture states that every $K_{t+1}$-minor-free graph is $t$-colourable. It is widely considered to be one of the most important conjectures in graph theory. If every $K_{t+1}$-minor-free graph has minimum degree at most $\delta$, then every $K_{t+1}$-minor-free graph is $(\delta+1)$-colourable by a minimum-degree-greedy algorithm. The purpose of this note is to prove a slightly better upper bound.