Independent Even Cycles in the Pancake Graph and Greedy Prefix-Reversal Gray Codes

Abstract

The Pancake graph $$P_n,\,n\geqslant 3$$Pn,nź3, is a Cayley graph on the symmetric group generated by prefix-reversals. It is known that $$P_n$$Pn contains any $$\ell $$l-cycle for $$6\leqslant \ell \leqslant n!$$6źlźn!. In this paper we construct a family of maximal (covering) sets of even independent (vertex-disjoint) cycles of lengths $$\ell $$l bounded by $$O(n^2)$$O(n2). We present the new concept of prefix-reversal Gray codes based on independent cycles which extends the known greedy prefix-reversal Gray code constructions given by Zaks and Williams. Cases of non-existence of codes based on the presented family of independent cycles are provided using the fastening cycle approach. We also give necessary condition for existence of greedy prefix-reversal Gray codes based on the independent cycles with arbitrary even lengths.