{
  "abstract": "A silver colouring is a proper colouring in which every colour appears in the closed neighbourhood of each vertex of a prescribed independent set. This local condition suffices when vertices are tested one at a time. We study a stronger requirement for simultaneous testing: whenever one or two vertices of the independent set are attacked together, each colour must supply distinct nearby defenders for them. We call this a pair-defensive silver colouring. For the hypercube $Q_d$, with one parity class as the attacked set, we prove that the maximum number of colours in a pair-defensive silver colouring is at most $\\lfloor(d+3)/2\\rfloor$, roughly half the ordinary silver-colouring target $d+1$. We construct colourings attaining this bound in four consecutive dimensions around every power of two, and we study the structure of the extremal, bound-attaining colourings. In each odd critical dimension, we characterize the extremal colourings by a partition of the defender parity into regular, triangle-free subgraphs of the halved cube. This characterization also has a local form in terms of coordinate matchings and a defect coordinate",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38993",
  "author_total": 2,
  "authors": [
    {
      "name": "Saman Farhat"
    },
    {
      "name": "Mehrak Shirkhani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dqcyehgahjzjb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38993",
  "fulltext_url": null,
  "id": "dqcyehgahjzjb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38993",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7215113972",
  "pdf_url": "https://arxiv.org/pdf/2609.38993",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dqcyehgahjzjb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38993",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38993"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Pair-Defensive Silver Colourings of Hypercubes",
  "title_text": "Pair-Defensive Silver Colourings of Hypercubes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/dqcyehgahjzjb",
  "venue": "arXiv",
  "versions": []
}