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 vertextransitive. In this paper, we show that BPn has every cycle of length ℓ with 8 ≤ ℓ ≤ 2 n n!. 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.