Isomorphism Classes of Bipartite Cycle Permutation Graphs

In this paper, we count the number of isomorphism classes of bipartite \(n\)-cyclic permutation graphs up to positive natural isomorphism and show that it is equal to the number of double cosets of the dihedral group \(D_n\) in the subgroup \(B_n\) of the symmetric group \(S_n\), consisting of parity-preserving or parity-reversing permutations.