{
  "abstract": "For a finite graph $H$, a connectivity graph-code is a family $\\mathcal C\\subseteq 2^{E(H)}$ such that $A\\triangle B$ is a connected spanning subgraph of $H$ whenever $A$ and $B$ are distinct members of $\\mathcal C$. Let $m(H)$ denote the maximum size of such a family, and let $f(d)$ be the largest integer $q$ for which $m(H)=q$ for infinitely many pairwise nonisomorphic $d$-regular graphs $H$. Restricting codewords to the edges incident with a vertex gives $f(d)\\leq 2^d$. Alon proved equality for all sufficiently large $d$ and asked whether it holds for every $d\\geq 4$. We answer this question affirmatively. More precisely, for every $d\\geq 4$ we construct infinitely many finite simple $d$-regular bipartite graphs carrying a linear connectivity graph-code of dimension $d$. The construction begins with a vector-labelled copy of $K_{d,d}$. For $d\\geq 7$, the required labelling follows from a probabilistic count over an irreducible conjugacy class in $\\mathrm{GL}_d(2)$; explicit matrices, verified by a short exact exhaustive program, cover $d=4,5,6$. Cyclic voltage lifts then produce the required infinite families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02953",
  "author_total": 1,
  "authors": [
    {
      "name": "Chenxiao Tian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3ss54pnn7xr4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02953",
  "fulltext_url": null,
  "id": "e3ss54pnn7xr4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02953",
  "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": "W7208729761",
  "pdf_url": "https://arxiv.org/pdf/2609.02953",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e3ss54pnn7xr4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.02953",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02953"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Alon's Question on Connectivity Graph-Codes: $f(d)=2^d$ for Every $d\\geq 4$",
  "title_text": "Alon's Question on Connectivity Graph-Codes: f(d)=2^d for Every d≥ 4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/e3ss54pnn7xr4",
  "venue": "arXiv",
  "versions": []
}