Bounds on the genus for 2-cell embeddings of prefix-reversal graphs

06/20/2023

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by   Saúl A. Blanco, et al.

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In this paper, we provide bounds for the genus of the pancake graph ℙ_n, burnt pancake graph 𝔹ℙ_n, and undirected generalized pancake graph ℙ_m(n). Our upper bound for 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 ℙ_n and 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.