bibkey: boyerkuenzellylepellico2025wellhued
authors: Geoffrey Boyer, Kirsti Kuenzel, Jeremy Lyle, Ryan Pellico
year: 2025
title: "Well-hued graphs with first difference two"
doi: 10.48550/arXiv.2506.04993
url: https://arxiv.org/abs/2506.04993v1
claim: "Section 3, Conjecture 2 proposes the unrestricted balanced multipartite characterization with independence number alpha(G)."
strata_touched:
  - D5/S3/Combinatorics/Graph/Products/BoyerKuenzelLylePellicoConjectureTwoRefutation
license: citation-only
triage: anchor

The unrestricted well-hued graph conjecture

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DOI: https://doi.org/10.48550/arXiv.2506.04993

Version 1: https://arxiv.org/abs/2506.04993v1

Section 3, Conjecture 2, printed page 10, is the target. The introduction defines maximal colorability using induced subgraphs and inclusion-maximal vertex sets, with recoloring allowed. Well-huedness concerns every positive number of colors.

The conjecture equates connected well-huedness, sequence (alpha(G), 2 alpha(G), ..., k alpha(G)), and chromatic number k with being a spanning subgraph of the balanced complete k-partite graph whose parts have size alpha(G), subject to an induced edgeless-graph exclusion outside every closed neighborhood. The overline on the forbidden K_alpha(G) is essential: it means an independent set of that order.

Theorem 4 treats independence number two. The paragraph introducing Conjecture 2 explicitly replaces two by alpha(G); the following paragraph returns to independence number two. The general conjecture is therefore distinct from that restricted theorem.

The Lean result refutes the general conjecture using the three by three rook graph. The known rook-graph classification and maximal-colorability obstruction belong to Goddard, Kuenzel and Melville (2022), cited in the companion Library note. No new classification or repaired conjecture is asserted.

The arXiv version page identifies the article license as CC BY-NC-SA 4.0 (https://creativecommons.org/licenses/by-nc-sa/4.0/). This note is an original summary for citation; no source code or proof text is copied.