Lengths of Cycles in Generalized Pancake Graphs

In this paper, we consider the lengths of cycles that can be embedded on the edges of the generalized pancake graphs which are the Cayley graph of the generalized symmetric group $S(m,n)$, generated by prefix reversals. The generalized symmetric group $S(m,n)$ is the wreath product of the cyclic group of order $m$ and the symmetric group of order $n!$. Our main focus is the underlying \emph{undirected} graphs, denoted by $\mathbb{P}_m(n)$. In the cases when the cyclic group has one or two elements, these graphs are isomorphic to the pancake graphs and burnt pancake graphs, respectively. We prove that when the cyclic group has three elements, $\mathbb{P}_3(n)$ has cycles of all possible lengths, thus resembling a similar property of pancake graphs and burnt pancake graphs. Moreover, $\mathbb{P}_4(n)$ has all the even-length cycles. We utilize these results as base cases and show that if $m>2$ is even, $\mathbb{P}_m(n)$ has all cycles of even length starting from its girth to a Hamiltonian cycle. Moreover, when $m>2$ is odd, $\mathbb{P}_m(n)$ has cycles of all lengths starting from its girth to a Hamiltonian cycle. We furthermore show that the girth of $\mathbb{P}_m(n)$ is $\min\{m,6\}$ if $m\geq3$, thus complementing the known results for $m=1,2.$