Cycles in the burnt pancake graphs [article]

Saúl A. Blanco, Charles Buehrle, Akshay Patidar
2019 arXiv   pre-print
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 BP_n, which is the Cayley graph of the group of signed permutations B_n using prefix reversals as generators, has similar properties. Indeed, BP_n is
more » ... -regular and vertex-transitive. In this paper, we show that BP_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 BP_n. We also present a complete characterization of all the 8-cycles in BP_n for n ≥ 2, which are the smallest cycles embeddable in BP_n, by presenting their canonical forms as products of the prefix reversal generators.
arXiv:1808.04890v2 fatcat:ujobwmm7orgblnwkpurq4vimci