Cycles in the burnt pancake graph

The pancake graph Pn is the Cayley graph of the symmetric group Sn on n elements generated by prefix reversals. Pn has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is (n−1)-regular, vertex-transitive, and one can embed cycles in it of length ℓ with 6≤ℓ≤n!. The burnt pancake graph BPn, which is the Cayley graph of the group of signed permutations Bn using prefix reversals as generators, has similar properties. Indeed, BPn is n-regular and vertex-transitive. In this paper, we show that BPn has every cycle of length ℓ with 8≤ℓ≤2nn!. The proof given is a constructive one that utilizes the recursive structure of BPn. We also present a complete characterization of all the 8-cycles in BPn for n≥2, which are the smallest cycles embeddable in BPn, by presenting their canonical forms as products of the prefix reversal generators.