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 \(s \le 2\) . 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.
