{
  "abstract": "For a graph $G$, let $f_t(G)$ denote the minimum of the maximum degree of an induced subgraph with $α(G)+t$ vertices, where $α(G)$ is the independence number, and write $f(G)=f_1(G)$. Huang's theorem gives $f(Q_k)\\ge\\lceil\\sqrt{k}\\rceil$ for the $k$-dimensional hypercube $Q_k$. We extend this lower bound to Cartesian products of $k$ bipartite graphs with perfect matchings and prove that equality holds when the factors are connected and each has a matched bridge. In particular, we prove that $f(T_1\\Box\\cdots\\Box T_k)=\\lceil\\sqrt{k}\\rceil$ whenever each $T_i$ is a tree with a perfect matching. We determine the sensitivity of every Cartesian product of paths, settling the even-path case left open by Zeng and Hou [J. Graph Theory 107 (2024), 169--180]. For these tree products, with $D=\\lceil\\sqrt{k}\\rceil$, we also prove that $f_t(T_1\\Box\\cdots\\Box T_k)=D$ whenever $1\\le t\\le 2^{D-\\lceil\\log_2D\\rceil-1}$. When $t=2$, this equality holds for all $k\\ge 2$, provided that at least one factor is not $K_2$. Matching cuts give an additional exact range for products of even-order paths. Finally, we prove that $f_2(Q_k)=\\lceil\\sqrt{k}\\rceil$ for every $k\\ge 2$ with $k\\not\\in \\{4,9\\}$, whereas $f_2(Q_4)=3$ and $3\\le f_2(Q_9)\\le 4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09483",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhen-Mu Hong"
    },
    {
      "name": "Zi-Yi Wu"
    },
    {
      "name": "Zheng-Jiang Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/72xfkm5v7m8rw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09483",
  "fulltext_url": null,
  "id": "72xfkm5v7m8rw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09483",
  "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": "W7220948195",
  "pdf_url": "https://arxiv.org/pdf/2610.09483",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/72xfkm5v7m8rw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09483",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09483"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graph Sensitivity of Cartesian Products with Matched Bridges",
  "title_text": "Graph Sensitivity of Cartesian Products with Matched Bridges",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/72xfkm5v7m8rw",
  "venue": "arXiv",
  "versions": []
}