{
  "abstract": "A \\emph{rainbow cycle} in a list-edge-colored graph is a cycle where no color appears more than once. We give a short and simple proof of a result of [ABSZZ] showing that a properly edge-colored $n$-vertex graph with average degree $Ω(\\log n\\log\\log n)$ always contains a rainbow cycle, and generalize it to weighted graphs with edges list colored with list size at most $\\ell$ and where the total weight of all edges containing a given color $i$ incident to a vertex is at most $Δ$, showing that $Ω_\\ell(Δ\\log n\\log\\log n)$ average degree guarantees a rainbow cycle. Using the reduction of [AG26] and the new generalized rainbow cycle theorem, we prove that for $r\\geq 3$, every binary linear $r$-query locally correctable code $E:\\mathbb{F}_2^k\\rightarrow \\mathbb{F}_2^n$ satisfies $k\\leq \\tilde{O}(n^{1-2/(r-1)})$ by showing a rainbow analogue of the odd $r$-LDC lower bounds of [BHKL25,JM25], closing the gap between the best known lower bounds for odd- and even-query binary linear $r$-LCCs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05034",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew D. Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d88n62xgh9vv3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05034",
  "fulltext_url": null,
  "id": "d88n62xgh9vv3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05034",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05034",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d88n62xgh9vv3.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.05034",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05034"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Short Proof and Generalization of the Rainbow Cycle Theorem and Improved Bounds for Binary Even-Query LCCs",
  "title_text": "A Short Proof and Generalization of the Rainbow Cycle Theorem and Improved Bounds for Binary Even-Query LCCs",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/d88n62xgh9vv3",
  "venue": "arXiv",
  "versions": []
}