{
  "abstract": "Let $C_n^k$ denote the $k$th power of the cycle $C_n$. The equivalence covering number $\\operatorname{eq}(G)$, studied by Alon, is the minimum number of equivalence subgraphs needed to cover all edges of $G$. King conjectured that $\\operatorname{eq}(C_n^k)=Ω(k)$ for sufficiently large $n$ relative to $k$. We study $\\operatorname{eq}(C_n^k)$ and prove that, for every $n \\geq 2k + 2$, we have $1 + \\lceil\\log_2(k+1)\\rceil \\leq \\operatorname{eq}(C_n^k) \\leq 1 + 2\\lceil\\log_2(k+1)\\rceil$. We further show that, whenever $n\\ge(k+1)\\max\\{41,5\\lceil\\log_2(k+1)\\rceil\\}$, \\begin{equation*} \\operatorname{eq}(C_n^k)\\in\\left\\{1+\\lceil\\log_2(k+1)\\rceil,\\,2+\\lceil\\log_2(k+1)\\rceil\\right\\}. \\end{equation*} Moreover, for every integer $n\\ge \\max\\{2k+2,k^2(k+2)\\}$, the following exact values hold in the indicated cases: \\begin{equation*} \\operatorname{eq}(C_n^k)=\\begin{cases} 1+\\lceil\\log_2(k+1)\\rceil, & k+1\\neq2^{\\lceil\\log_2(k+1)\\rceil} \\text{ or }(k+1)\\mid n,\\\\[3pt] 2+\\lceil\\log_2(k+1)\\rceil, & k+1=2^{\\lceil\\log_2(k+1)\\rceil} \\text{ and }2\\nmid n. \\end{cases} \\end{equation*}",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15635",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhanping Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eegk1fgssbxkp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15635",
  "fulltext_url": null,
  "id": "eegk1fgssbxkp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15635",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221783540",
  "pdf_url": "https://arxiv.org/pdf/2610.15635",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eegk1fgssbxkp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15635",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15635"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Covering powers of cycles with equivalence subgraphs",
  "title_text": "Covering powers of cycles with equivalence subgraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/eegk1fgssbxkp",
  "venue": "arXiv",
  "versions": []
}