  In this paper, we provide bounds for the genus of the pancake graph
$\mathbb{P}_n$, burnt pancake graph $\mathbb{BP}_n$, and undirected generalized
pancake graph $\mathbb{P}_m(n)$. Our upper bound for $\mathbb{P}_n$ is sharper
than the previously-known bound, and the other bounds presented are the first
of their kind. Our proofs are constructive and rely on finding an appropriate
rotation system (also referred to in the literature as Edmonds' permutation
technique) where certain cycles in the graphs we consider become boundaries of
regions of a 2-cell embedding. A key ingredient in the proof of our bounds for
the genus $\mathbb{P}_n$ and $\mathbb{BP}_n$ is a labeling algorithm of their
vertices that allows us to implement rotation systems to bound the number of
regions of a 2-cell embedding of said graphs.
