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

  We study the invariant p(G), the minimum number of palette expansions over all vertex orderings of a greedy coloring,
  and prove that χ(G) = 1 + p(G). Our main contribution is the Patio Adjacency Lemma (Theorem 3.4): in any optimal
  greedy palette-expansion coloring with expansion centers c₁,…,cₖ, the center cⱼ has, for every i < j, a neighbor in
  color class Aᵢ; hence every pair of color classes is joined by an edge with one endpoint at an expansion center. The
  lemma is proved and verified computationally on 130,000+ graphs.

  We then study a branch-set construction seeded by the color classes. The Patio Adjacency Lemma makes the initial
  configuration pairwise-adjacent at no cost; the construction maintains adjacency while attempting to repair
  connectivity. We prove what it guarantees unconditionally (disjointness, an adjacency invariant, and a bounded
  connectivity potential), and we are explicit that producing internally connected branch sets — equivalently,
  partitioning V(G) into k connected, pairwise-adjacent sets — is exactly Hadwiger's conjecture for that k (Open Problem
  6.1), which we do not settle. On the tested graphs the construction achieved the pairwise-adjacency half universally;
  internal connectivity was generally not achieved. We make no claim about Hadwiger's conjecture for k ≥ 7.

  Changes in v24: corrected the Section 4 construction to be internally consistent; proved the connectivity-potential
  lemma in full; corrected an open problem that was in fact immediate; and revised Section 5 to report honestly that the
  experiments confirm the adjacency half, not full Kₖ-minor certificates.

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

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