{
  "abstract": "A bicirculant is a regular graph that admits an automorphism having two vertex-orbits of the same size. A bicirculant can be described as follows. Given an integer $m \\ge 1$ and sets $R, S, T \\subseteq \\mathbb{Z}_m$ such that $R=-R$, $T=-T$, $0 \\not\\in R \\cup T$ and $0 \\in S$, the graph $B(m;R,S,T)$ has vertex set $V=\\{u_0,\\dots,u_{m-1},v_0,\\dots,v_{m-1}\\}$ and edge set $E=\\{u_iu_{i+j}| \\ i \\in \\mathbb{Z}_m, j \\in R\\} \\cup \\{v_iv_{i+j}| \\ i \\in \\mathbb{Z}_m, j \\in T\\} \\cup \\{u_iv_{i+j}| \\ i \\in \\mathbb{Z}_m, j \\in S\\}.$ Some of the well known bicirculants include the generalized Petersen graphs and Cayley graphs of dihedral groups. In 1983, Brian Alspach proved that the only non-hamiltonian generalized Petersen graphs are $G(m,2)$ where $m \\equiv 5 \\pmod 6$. In a recent paper we conjectured that this is the only exception among regular connected bicirculants of degree at least two, and we have verified the conjecture for bicirculants with $|S|\\le 3$. Results of Alspach, Chen and Dean imply that the conjecture also holds for all bicirculants with $m$ even and $\\gcd(m,S)=1$. In this paper, we show that the conjecture holds for all bicirculants with $|S| \\geq 4$ and $\\gcd(m,S)>1$ such that $R\\cup T$ contains an element that is not coprime to $\\gcd(m,S)$ and, if $m$ is odd, is also not divisibile by $\\gcd(m, S)$ or $R \\cap T$ contains an element that is coprime to $\\gcd(m,S)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05531",
  "author_total": 3,
  "authors": [
    {
      "name": "Simona Bonvicini"
    },
    {
      "name": "Tomaž Pisanski"
    },
    {
      "name": "Arjana Žitnik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fwsbkcv6nzcrd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05531",
  "fulltext_url": null,
  "id": "fwsbkcv6nzcrd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05531",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05531",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fwsbkcv6nzcrd.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05531",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05531"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hamilton cycles in bicirculants with non-coprime connection sets",
  "title_text": "Hamilton cycles in bicirculants with non-coprime connection sets",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/fwsbkcv6nzcrd",
  "venue": "arXiv",
  "versions": []
}