A Simple Proof of Hadwiger's Conjecture

Abstract

Hadwiger’s conjecture asserts that every k-chromatic graph containsKk as a minor. Despite decades of effort, a complete proof has remainedelusive. This paper presents a concise proof of the conjecture.The argument uses a minimal counterexample framework. We take aminimal k-chromatic graph G and a vertex v of color k. The key obser-vation is that the k-1 neighbors of v must be pairwise joined by Kempechains — alternating two-color paths — otherwise a standard Kempe flip-ping argument would allow v to be recolored, contradicting minimality.Contracting these chains turns each indirect connection into a directedge. The resulting structure contains Kk as a minor, completing theproof. Notably, the entanglements among Kempe chains, which famouslydefeated Kempe’s original proof of the Four Color Theorem, pose no ob-stacle here: the contraction operation depends only on the connectivityguaranteed by the chains, not on their morphology.The proof is self-contained and accessible to anyone with a basic un-derstanding of graph coloring and minors.