All rose window graphs are hamiltonian

Abstract: A bicirculant is a regular graph that admits an automorphism having two 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 $E={u_iu_{i+j}| \ i \in \mathbb{Z}m, j \in R} \cup {v_iv{i+j}| \ i \in \mathbb{Z}m, j \in T} \cup {u_iv{i+j}| \ i \in \mathbb{Z}_m, j \in S}.$ Let $m \ge 3$ be a positive integer and $a,b,c \in \mathbb{Z}_m \setminus{0}$ with $a,b \ne m/2$. If we take $R = {a,-a}$, $S = {0,c}$ and $T = { b, -b}$, the graph $B(m;R,S,T)$ is a generalized rose window graph. A rose window graph has the additional property that at least one of $a,b$ is relatively prime to $m$. In this paper we show that all generalized rose window graphs are hamiltonian. As a consequence we obtain that every connected bicirculant $B(m;R,S,T)$ with $|S| \ge 3$ is hamiltonian if $m$ is a product of at most three prime powers. In particular, every connected bicirculant $B(m;R,S,T)$ with$|S| \ge 3$ is hamiltonian for even $m<210$ and odd $m < 1155$.