The orbits of twisted cubes

Two vertices $u$ and $v$ in a graph $G=(V,E)$ are in the same orbit if there exists an automorphism $\varphi$ of $G$ such that $\varphi \left(u\right)=v$. The orbit number of a graph $G$, denoted by $Orb\left(G\right)$, is the smallest number of orbits that partition $V\left(G\right)$. All vertex-transitive graphs $G$ satisfy $Orb\left(G\right)=1$. Since the $n$-dimensional hypercube, denoted by $Q_n$, is vertex-transitive, it follows that $Orb\left(Q_n\right)=1$ for $n\ge 1$. The twisted cube (Abraham and Padmanabhan, 1991), denoted by $T{Q}_n$, is an interesting variant of the hypercube. In this paper, we prove that $Orb\left(T{Q}_n\right)=1$ if $n\le 4$ and $Orb\left(T{Q}_n\right)=2^{\lfloor \frac{n-3}{2}\rfloor }$ if $n\ge 5$.