On the Hamiltonian Bicirculants

TL;DR

This paper investigates Hamiltonian cycles in bicirculant graphs, verifying a conjecture for certain cases and establishing conditions under which these graphs are Hamiltonian, thus advancing understanding of their structural properties.

Contribution

The paper develops new tools to partially verify a conjecture about Hamiltonicity in bicirculants, extending known results to broader classes of these graphs.

Findings

Conjecture holds for bicirculants with s ≤ 2.

Bicirculants with s ≥ 3 are Hamiltonian if m has up to three prime factors.

All bicirculants with d-s odd are Hamiltonian.

Abstract

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.