{
  "abstract": "Word-representable graphs are a class of graphs that can be represented by words, where edges and non-edges are determined by the alternation of letters in those words. The tensor product $G \\times H$ (also known as the direct product or Kronecker product) is one of the four standard graph products. Problem 7.2.5 in Kitaev and Lozin's book \\emph{Words and Graphs} (Springer, 2015) raised three open questions regarding the word-representability of tensor products. Despite extensive research on word-representable graphs, the word-representability of tensor products seems to have received no attention. This paper not only answers 2.5 of the questions posed by Kitaev and Lozin in Problem 7.2.5, but also initiates a systematic study, focusing on four fundamental families: wheel graphs $W_n$, complete graphs $K_n$, the Mycielskian of the cycle graph $μ_n$, and the extended Mycielskian of the cycle graph $μ'_n$. Among our main results, we prove that $W_{2n} \\times G$, $μ_{2n} \\times G$, and $μ'_{2n} \\times G$ are always word-representable for any graph $G$; that $K_n \\times K_m$ is word-representable if and only if $\\min\\{n,m\\} \\leq 3$; and that tensor products $G \\times H$ containing $W_{2n+1} \\times W_{2m+1}$ or $μ_{2n+1} \\times μ_{2m+1}$ or $μ'_{2n+1} \\times μ'_{2m+1}$ as induced subgraphs are non-word-representable. Our proofs exploit the hereditary nature of non-word-representability and the presence of non-comparability neighbourhoods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20881",
  "author_total": 1,
  "authors": [
    {
      "name": "Nawaf Alshammari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ek1hgwb99f7gh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20881",
  "fulltext_url": null,
  "id": "ek1hgwb99f7gh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20881",
  "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|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213939915",
  "pdf_url": "https://arxiv.org/pdf/2609.20881",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ek1hgwb99f7gh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.20881",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20881"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Word-Representability of Tensor Product Graphs",
  "title_text": "On the Word-Representability of Tensor Product Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/ek1hgwb99f7gh",
  "venue": "arXiv",
  "versions": []
}