{
  "abstract": "Let $Γ=(V,E)$ be a graph and let $a,b$ be nonnegative integers. A nonempty proper subset $C\\subset V$ is an \\emph{$(a,b)$-regular set} if each vertex of $C$ has exactly $a$ neighbors in $C$ and each vertex outside $C$ has exactly $b$ neighbors in $C$. Perfect codes and total perfect codes are the $(0,1)$- and $(1,1)$-regular sets. For every connected circulant quartic graph $\\operatorname{Cay}(\\mathbb{Z}_n,\\{x^{\\pm k},x^{\\pm\\ell}\\})$ and every pair $(a,b)$ we decide, in terms of $n$, $k$ and $\\ell$, whether an $(a,b)$-regular set exists. Existence is proved by exhibiting unions of residue classes; non-existence comes from a Fourier-analytic obstruction combined with the fact that certain sums of roots of unity cannot vanish.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09414",
  "author_total": 6,
  "authors": [
    {
      "name": "A. Abdollahi"
    },
    {
      "name": "J. Bagherian"
    },
    {
      "name": "F. Jafari"
    },
    {
      "name": "M. Khatami"
    },
    {
      "name": "Z. Shokoohi"
    },
    {
      "name": "R. Sobhani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86zh5wp29n6d2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09414",
  "fulltext_url": null,
  "id": "86zh5wp29n6d2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09414",
  "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|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7212111635",
  "pdf_url": "https://arxiv.org/pdf/2609.09414",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86zh5wp29n6d2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.09414",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09414"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Existence of regular sets in circulant quartic graphs: a complete characterization",
  "title_text": "Existence of regular sets in circulant quartic graphs: a complete characterization",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/86zh5wp29n6d2",
  "venue": "arXiv",
  "versions": []
}