bibkey: lalou2026completely authors: Mohammed Lalou; Nader Mbarek; Abdallah Skender; Olivier Togni year: 2026 title: “Completely Independent Spanning Trees in Split Graphs: Structural Properties and Complexity” doi: 10.48550/arXiv.2512.15486 url: https://arxiv.org/abs/2512.15486v2 claim: “Conjecture 1 asserts that every hypergraph satisfies chi_p^2(H) = chi_p(H) - ceil(alpha_{chi_p(H)}(H)/2).” strata_touched:

• D5/S3/ConceptDynamics/GraphColoring/PanchromaticPairingConjectureRefutation license: CC-BY-4.0 triage: anchor

Panchromatic and bipanchromatic coloring conjecture

Version 2 of the source was checked at the stable arXiv locator below. Section 2 defines a panchromatic k-coloring by requiring every hyperedge to contain every color. A unique color occurs once in the whole hypergraph. A bipanchromatic coloring is panchromatic and has no unique color, equivalently every global color class has at least two vertices. The panchromatic and bipanchromatic numbers are maxima over feasible color counts.

The Section 3 preamble paragraph immediately preceding Section 3.1 defines alpha_k as the minimum number of unique colors among all panchromatic k-colorings. Section 5, Equation (5.1), and Conjecture 1 state:

Every hypergraph H satisfies chi_p^2(H) = chi_p(H) - ceil(alpha_{chi_p(H)}(H)/2).

The source reports positive integer-programming tests before proposing the conjecture. It does not contain or attest the repository’s six-vertex counterexample. The repository result is therefore recorded as repo-derived, with no world-priority claim.

A bounded screen of the primary and latest arXiv records, arXiv results for bipanchromatic coloring, author and OpenAlex records, the project library, pinned Mathlib, and public Lean keyword results found no exact earlier resolution in the searched scope. This is not a worldwide novelty or priority claim.

Crossref records the journal publication as Discrete Applied Mathematics 391 (2026), pages 511–522, DOI 10.1016/j.dam.2026.05.028. The publisher full text remains unverified because access returned HTTP 403; the verified locator below therefore remains the arXiv version used for the exact statement.

Verified locator

• DOI: 10.48550/arXiv.2512.15486
• URL: https://arxiv.org/abs/2512.15486v2
• Checked text: https://arxiv.org/html/2512.15486v2, dated 28 July 2026, especially Section 2 paragraphs 2–3, the Section 3 preamble paragraph immediately preceding Section 3.1 (https://arxiv.org/html/2512.15486v2#S3.p6), and Section 5 Conjecture 1. The checked local HTML has SHA-256 822afac3cd61ccc97ca2f75cf8ba236538793e54c8b3b21cc7b3e81acb0c776a.