Cycles in the burnt pancake graphs
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The pancake graph of S_n, the symmetric group on n elements, has been shown to have many interesting properties that makes it a useful network scheme for parallel processors. For example, it is (n-1)-regular, vertex-transitive, and pancyclic (one can find cycles of any length from its girth up to the number of vertices of the graph). The burnt pancake graph BP_n, which is obtained as the Cayley graph of the group B_n of signed permutations on n elements using prefix reversal as generators, has similar properties. Indeed, BP_n is n-regular and vertex-transitive. In this paper, we show that BP_n is also pancyclic. Our proof is a recursive construction of the cycles. We also present a complete characterization of all the 8-cycles in BP_n for n ≥ 2 by presenting their canonical forms as products of the prefix reversal generators.
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