{
  "abstract": "A majority C-colouring partitions a graph into classes in which every vertex has at least half of its neighbours. Write $M(G)$ for the maximum number of classes. We determine $M(K_q^{\\square(2k+1)})=\\left\\lfloor\\frac{q^{k+1}}{\\lfloor q/2\\rfloor+1}\\right\\rfloor\\qquad(q\\ge3,\\ k\\ge0).$ The lower bound follows from explicit rectangular partitions and a uniform three-dimensional bridge. A punctured-rectangle construction supplies both the bridge and a partition in the intermediate dimension. The upper bound is a classical consequence of Hamming edge isoperimetry; an elementary proof is included. We also prove $M(C_m\\square P_n)=\\frac n2\\lfloor m/2\\rfloor$ for $m\\ge4$ and even $n\\ge2$, contradicting the cylinder assertion of Conjecture 4 in arXiv:2608.27669v1 for odd $m\\ge7$ and even $n\\ge6$. Further layer bounds determine additional cylinder and torus families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31674",
  "author_total": 1,
  "authors": [
    {
      "name": "ZeHua Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4m1freaghbmms.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31674",
  "fulltext_url": null,
  "id": "4m1freaghbmms",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31674",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214748646",
  "pdf_url": "https://arxiv.org/pdf/2609.31674",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4m1freaghbmms.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31674",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31674"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact majority C-colourings of balanced Hamming graphs and grids",
  "title_text": "Exact majority C-colourings of balanced Hamming graphs and grids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4m1freaghbmms",
  "venue": "arXiv",
  "versions": []
}