{
  "abstract": "We determine weak $k$-metric dimensions for several families of Hamming graphs. For $n\\ge3$, $m>n$, and $3\\le k\\le2n$, a cyclic construction proves Conjecture 6.1 of Fernández, Klavžar, Kuziak, Muñoz-Márquez and Yero (2026) on $K_n\\square K_m$. For these rectangular products, we also prove that $\\operatorname{wdim}_2(K_n\\square K_m)=m$ exactly when $m\\ge2n-2$. For hypercubes $Q_d$ with $d\\ge2$, we show that consecutive parameters $2s-1$ and $2s$ have identical weak resolving sets. Near the maximum parameter, a reduction to restricted-distance binary codes determines $\\operatorname{wdim}_{2^d-t}(Q_d)$ for every feasible deficit $0\\le t\\le15$. For $L\\ge1$ and $t\\in\\{2L,2L+1\\}$, we prove the stabilization formula $\\operatorname{wdim}_{2^d-t}(Q_d)=2^d-L$ for $d\\ge2^L+1$, and show that this threshold is sharp. For each fixed $q\\ge3$ and deficit $t$, we also determine $\\operatorname{wdim}_{2q^{d-1}-t}(K_q^{\\square d})$ in all sufficiently large dimensions, with an explicit sufficient condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32161",
  "author_total": 1,
  "authors": [
    {
      "name": "Aryan Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/559h83y2vka46.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32161",
  "fulltext_url": null,
  "id": "559h83y2vka46",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32161",
  "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|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7214716139",
  "pdf_url": "https://arxiv.org/pdf/2609.32161",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/559h83y2vka46.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32161",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32161"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Weak $k$-metric dimension of Hamming graphs: rectangular products and near-maximum parameters",
  "title_text": "Weak k-metric dimension of Hamming graphs: rectangular products and near-maximum parameters",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/559h83y2vka46",
  "venue": "arXiv",
  "versions": []
}