The Pancake graphis the Cayley graph on the symmetric group of permutations of {1,2,…,n} with the generating set of prefix-reversals. By today two Hamiltonian cycles` constructions are known: the first was given by S. Zaks in 1984 in terms of suffix-reversals,the second was suggested by A. Williams and J. Sawada in 2013 along with the general construction of greedy Prefix-reversal Gray codes. 

In this talk we show that both constructions are based on the maximal sets of independent cycles. We define the family of maximal sets of independent cycles of even length in the Pancake graph, present the definition of the Hamiltonian cyclebased on independent cycles in this graph and study the existence of such Hamiltonian cycles. We also show that greedy Prefix-reversal Gray codes are characterized by our construction and provide the necessary condition of existence of these cycles.

At the end we present the open questions for the related research.

This work is joint work with Elena Konstantinova (Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russia).