A note on Hadwiger's conjecture: Another proof that every 4-chromatic graph has a $K_4$ minor

TL;DR

This paper presents a new proof confirming that any graph with chromatic number at least 4 necessarily contains a $K_4$ minor, supporting Hadwiger's Conjecture for this case.

Contribution

It introduces a novel proof technique that derives the $K_4$ minor from a proper 3-coloring of a subgraph, offering an alternative approach to the conjecture.

Findings

Every 4-chromatic graph has a $K_4$ minor.

The proof uses proper 3-coloring of a subgraph.

Supports Hadwiger's Conjecture for the case of chromatic number 4.

Abstract

The first non-obvious case of Hadwiger's Conjecture states that every graph $G$ with chromatic number at least 4 has a $K_4$ minor. We give a new proof that derives the $K_4$ minor from a proper 3-coloring of a subgraph of $G$.