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.

翻译：本文给出了煎饼图$\mathbb{P}_n$、烧焦煎饼图$\mathbb{BP}_n$及无向广义煎饼图$\mathbb{P}_m(n)$的亏格界。其中$\mathbb{P}_n$的上界优于已知结果，其余界均为首次提出。我们的证明是构造性的，依赖于寻找合适的旋转系统（文献中亦称为Edmonds置换技巧），使得所研究图中的特定环成为2-胞腔嵌入区域的边界。在证明$\mathbb{P}_n$和$\mathbb{BP}_n$亏格界的关键步骤中，我们提出了一种顶点标号算法，该算法使得我们能够通过实施旋转系统来界定这些图的2-胞腔嵌入的区域数量。