A Patio Adjacency Lemma for Greedy Colorings, with Computational Evidence Toward Branch-Set Connectivity

We introduce the palette-expansion number p(G) and give a short proof that χ(G) = 1 + p(G) for all connected simple graphs. Our main contribution is the Patio Adjacency Lemma: in any optimal greedy palette-expansion coloring with expansion centers c₁, …, cₖ, the center cⱼ has a neighbor in every color class Aᵢ for i < j. Verified computationally on 130,000+ graphs with zero failures. We also describe a hybrid branch-set construction tested on 562 graphs that produces Kₖ-minor certificates in all tested cases. The formal proof that this construction always succeeds is an open problem, stated precisely in the paper. Hadwiger's conjecture for k ≥ 7 remains open.

Code: https://github.com/mizaelantoniotovarreyes/chromatic-hadwiger

Project site: https://hadwiger.oryum.ai/