Cycles in the burnt pancake graph

Saúl A. Blanco, Charles Buehrle, Akshay Patidar

Abstract: The pancake graph P n is the Cayley graph of the symmetric group S n on n elements generated by prefix reversals. P n 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 B P n , which is the Cayley graph of the group of signed permutations B n using prefix reversals as generators, has similar properties. Indeed, B P n is n -regular and vertex-transitive. In this paper, we show that B P n 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 B P n . We also present a complete characterization of all the 8-cycles in B P n for n ≥ 2 , which are the smallest cycles embeddable in B P n , by presenting their canonical forms as products of the prefix reversal generators. Previous article in issue Next article in issue