On the Hamiltonian Bicirculants
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arXiv
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| Format: | Preprint |
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2025
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| author | Bonvicini, S. Pisanski, T. Žitnik, A. |
| author_facet | Bonvicini, S. Pisanski, T. Žitnik, A. |
| contents | A bicirculant is a regular graph that admits a semi-regular automorphism with two vertex-orbits of the same size. By $m$ we denote the size of vertex-orbits and by $d$ the valence of a bicirculant. Furthermore, we denote by $s$ the valence of the bipartite graph joining the two vertex-orbits. In 1983, Brian Alspach proved that the only non-hamiltonian generalized Petersen graphs are $G(m,2)$ with $m \equiv 5 \pmod 6$. In a recent paper we conjectured that this is the only exception among regular, connected bicirculants of degree $d > 1$ and we have verified the conjecture for the quartic bicirculants with $s=2$, also known as the generalized rose window graphs.
In this paper we develop tools and apply them for a partial verification of the conjecture. We show that the conjecture holds for all bicirculants with $s \leq 2$. As a consequence we obtain that every connected bicirculant with $s \ge 3$ is hamiltonian if $m$ is a product of at most three prime powers. In particular, every connected bicirculant with $s \ge 3$ is hamiltonian for even $m<210$ and odd $m < 1155$. Our results imply that many other families of bicirculants are hamiltonian. For example, all bicirculants with $d-s$ odd are hamiltonian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Hamiltonian Bicirculants Bonvicini, S. Pisanski, T. Žitnik, A. Combinatorics 05C45, 05C25, 05C76, 05C70, 05E18 A bicirculant is a regular graph that admits a semi-regular automorphism with two vertex-orbits of the same size. By $m$ we denote the size of vertex-orbits and by $d$ the valence of a bicirculant. Furthermore, we denote by $s$ the valence of the bipartite graph joining the two vertex-orbits. In 1983, Brian Alspach proved that the only non-hamiltonian generalized Petersen graphs are $G(m,2)$ with $m \equiv 5 \pmod 6$. In a recent paper we conjectured that this is the only exception among regular, connected bicirculants of degree $d > 1$ and we have verified the conjecture for the quartic bicirculants with $s=2$, also known as the generalized rose window graphs. In this paper we develop tools and apply them for a partial verification of the conjecture. We show that the conjecture holds for all bicirculants with $s \leq 2$. As a consequence we obtain that every connected bicirculant with $s \ge 3$ is hamiltonian if $m$ is a product of at most three prime powers. In particular, every connected bicirculant with $s \ge 3$ is hamiltonian for even $m<210$ and odd $m < 1155$. Our results imply that many other families of bicirculants are hamiltonian. For example, all bicirculants with $d-s$ odd are hamiltonian. |
| title | On the Hamiltonian Bicirculants |
| topic | Combinatorics 05C45, 05C25, 05C76, 05C70, 05E18 |
| url | https://arxiv.org/abs/2510.23420 |