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Hamilton cycles in bicirculants with non-coprime connection sets — arXiv:2611.05531

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.05531
License (as archived): CC0-1.0
Archived: 2026-11-05T18:28:21.630Z
SHA-256 of the archived text: 662a2734ba3096cfdd377975aaf6b6757c53c14da66c5d280e8687cb532852e5
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How to cite

Hamilton cycles in bicirculants with non-coprime connection sets — arXiv:2611.05531 — arXiv (en), https://arxiv.org/abs/2611.05531 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e9c8cc8dfd3e (Metraipolis archive dc_e9c8cc8dfd3e, sha256 662a2734ba3096cfdd377975aaf6b6757c53c14da66c5d280e8687cb532852e5)

BibTeX

@misc{dce9c8cc8dfd3e,
  title = {Hamilton cycles in bicirculants with non-coprime connection sets — arXiv:2611.05531},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.05531},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e9c8cc8dfd3e; sha256 662a2734ba3096cfdd377975aaf6b6757c53c14da66c5d280e8687cb532852e5},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.05531. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\gcd(m,S)$.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2611.05531 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-05T18:18:33.994Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).