Refutation of the panchromatic pairing conjecture
Abstract
A six-vertex hypergraph refutes the proposed equality between panchromatic and bipanchromatic extrema.
Definition 1.1 (Lalou–Mbarek–Skender–Togni Conjecture 1).
Formalization. D5/S3/ConceptDynamics/GraphColoring/PanchromaticPairingConjectureRefutation.claim (✓ std3).
Citation. Mohammed Lalou; Nader Mbarek; Abdallah Skender; Olivier Togni (2026). Completely Independent Spanning Trees in Split Graphs: Structural Properties and Complexity. DOI: 10.48550/arXiv.2512.15486. URL: https://arxiv.org/abs/2512.15486v2.
Commentary.
For a finite set-valued hypergraph, p and b are attained maxima over all panchromatic and bipanchromatic color counts. The value a is an attained minimum of the number of globally singleton color fibers over all maximizing p-colorings. The claim states b = p - ceil(a/2).
Theorem 1.2 (Conjecture 1 is false).
Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/GraphColoring/PanchromaticPairingConjectureRefutation.result (✓ std3). ∎
Resolves. Problems/lalou-mbarek-skender-togni-conjecture-one-refutation (refuted) by D5/S3/ConceptDynamics/GraphColoring/PanchromaticPairingConjectureRefutation.result.
Source. Repository-derived.
Acknowledgement. Mohammed Lalou; Nader Mbarek; Abdallah Skender; Olivier Togni (2026). Completely Independent Spanning Trees in Split Graphs: Structural Properties and Complexity. DOI: 10.48550/arXiv.2512.15486. URL: https://arxiv.org/abs/2512.15486v2.
Commentary.
On vertices 0 through 5, take the three edges {0,1,2,3}, {0,1,2,4}, and {0,1,2,5}. Their panchromatic maximum is four. Every maximizing coloring has the three core colors globally singleton and a common petal color, so the minimum singleton count is three. The bipanchromatic maximum is three. Thus the left side is 3 while the proposed right side is 4 - ceil(3/2) = 2.