{
  "abstract": "The following is a natural and fundamental question for a graph $G$: if an induced subgraph $H$ of $G$ has $x$ more vertices than a maximum independent set, what can be said about the maximum degree of $H$ as a function of $x$? The case $x=1$ is already of considerable interest. For example, in his celebrated proof of the sensitivity conjecture, Hao Huang showed that every induced subgraph of the hypercube $Q_n$ on more than $2^{n-1}$ vertices has maximum degree at least $\\sqrt{n}$. Chung, Fúredi, Graham, and Seymour proved that this bound is tight. In this paper, we study this question when $G$ is either the $n$-fold Hamming product or the $n$-fold tensor product of a triangle. For both graphs, we determine the exact minimum possible average degree of an induced subgraph of a prescribed size. We also prove that the tensor product exhibits several Huang-like phenomena. For the Hamming product, we show that, for several size densities and large $n$, the minimum possible maximum degree is asymptotically equal to the minimum possible average degree. Finally, we extend several of the results from the triangle to an arbitrary complete graph $K_k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32903",
  "author_total": 1,
  "authors": [
    {
      "name": "Elizaveta Popova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9sq7gbna7rcjy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32903",
  "fulltext_url": null,
  "id": "9sq7gbna7rcjy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32903",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214731599",
  "pdf_url": "https://arxiv.org/pdf/2609.32903",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9sq7gbna7rcjy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32903",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32903"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Minimum Possible Maximum Degree of Induced Subgraphs of Product Graphs",
  "title_text": "On the Minimum Possible Maximum Degree of Induced Subgraphs of Product Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9sq7gbna7rcjy",
  "venue": "arXiv",
  "versions": []
}