Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph

Abstract: The Bubble-sort graph $BS_n,\,n\geqslant 2$, is a Cayley graph over the symmetric group $Sym_n$ generated by transpositions from the set ${(1 2), (2 3),\ldots, (n-1 n)}$. It is a bipartite graph containing all even cycles of length $\ell$, where $4\leqslant \ell\leqslant n!$. We give an explicit combinatorial characterization of all its $4$- and $6$-cycles. Based on this characterization, we define generalized prisms in $BS_n,\,n\geqslant 5$, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.