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


The Pancake graph \(P_n,\,n\geqslant 3\), is a Cayley graph on the symmetric group generated by prefix-reversals. It is known that \(P_n\) contains any \(\ell \)-cycle for \(6\leqslant \ell \leqslant n!\). In this paper we construct a family of maximal (covering) sets of even independent (vertex-disjoint) cycles of lengths \(\ell \) bounded by \(O(n^2)\). 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.

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Correspondence to Alexey Medvedev.

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The research was supported by Grant 15-01-05867 of the Russian Foundation of Basic Research and by Grant NSh-1939.2014.1 of President of Russia for Leading Scientific Schools.

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Konstantinova, E., Medvedev, A. Independent Even Cycles in the Pancake Graph and Greedy Prefix-Reversal Gray Codes. Graphs and Combinatorics 32, 1965–1978 (2016).

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  • Pancake graph
  • Pancake sorting
  • Even cycles
  • Disjoint cycle cover
  • Prefix-reversal Gray codes
  • Fastening cycle

Mathematics Subject Classification

  • 05C45
  • 05C25
  • 05C38
  • 90B10