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

## TL;DR

This paper characterizes small cycles in the Bubble-sort graph, introduces generalized prisms, and proposes a new method for constructing Hamiltonian cycles in these graphs.

## Contribution

It provides an explicit combinatorial characterization of 4- and 6-cycles and introduces generalized prisms to construct Hamiltonian cycles in Bubble-sort graphs.

## Key findings

- Explicit characterization of 4- and 6-cycles in BS_n
- Definition of generalized prisms in BS_n
- New approach to construct Hamiltonian cycles

## 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.