The Orbits of Folded Crossed Cubes.

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. [ABSTRACT FROM AUTHOR]