The Orbits of Folded Crossed Cubes

Abstract

Abstract Two vertices $u$ and $v$ in a graph $G=(V,E)$ are in the same orbit if there exists an automorphism $\phi $ of $G$ such that $\phi (u)=v$. The orbit number of a graph $G$, denoted by $Orb(G)$, is the smallest number of orbits, which form a partition of $V(G)$, in $G$. All vertex-transitive graphs $G$ are with $Orb(G)=1$. Since the $n$-dimensional hypercube, denoted by $Q_{n}$, is vertex-transitive, it follows that $Orb(Q_{n})=1$ for $n\geq 1$. Pai, Chang, and Yang proved that the $n$-dimensional folded crossed cube, denoted by $FCQ_{n}$, is vertex-transitive if and only if $n\in \{1,2,4\}$, namely $Orb(FCQ_{1})=Orb(FCQ_{2})=Orb(FCQ_{4})=1$. In this paper, we prove that $Orb(FCQ_{n})=2^{\lceil \frac{n}{2}\rceil -2}$ if $n\geq 6$ is even and $Orb(FCQ_{n}) = 2^{\lceil \frac{n}{2}\rceil -1}$ if $n\geq 3$ is odd.