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

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Abstract

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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References

  1. Gilbert, E.N.: Gray codes and paths on the \(n\)-cube. Bell Syst. Tech. J. 37, 815–826 (1958)

    Article  MathSciNet  Google Scholar 

  2. Joichi, J.T., White, D.E., Williamson, S.G.: Combinatorial Gray codes. SIAM J. Comput. 9(1), 130–141 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kanevsky, A., Feng, C.: On the embedding of cycles in Pancake graphs. Parallel Comput. 21, 923–936 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Konstantinova, E.V., Medvedev, A.N.: Cycles of length seven in the Pancake graph. Diskretn. Anal. Issled. Oper. 17(5), 46–55 (2010). (in Russian)

    MathSciNet  MATH  Google Scholar 

  5. Konstantinova, E., Medvedev, A.: Small cycles in the Pancake graph. Ars Mathematica Contemporanea 7, 237–246 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Savage, C.: A survey of combinatorial Gray codes. SIAM Rev. 39, 605–629 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Sheu, J.J., Tan, J.J.M., Hsu, L.H., Lin, M.Y.: On the cycle embedding of Pancake graphs. In: Proceedings of 1999 National Computer Symposium, pp. C414–C419 (1999)

  8. Sheu, J.J., Tan, J.J.M., Chu, K.T.: Cycle embedding in pancake interconnection networks. In: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory in Taiwan, pp. 85–92 (2006)

  9. Skiena, S.: Hypercubes, Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, pp. 148–150. Addison–Wesley, Reading (1990)

    Google Scholar 

  10. Williams, A.: The greedy gray code algorithm. LNCS 8037, 525–536 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Williams, A., Sawada, J.: Greedy Pancake flipping. Electron. Notes Discrete Math. 44, 357–362 (2013)

    Article  Google Scholar 

  12. Zaks, S.: A new algorithm for generation of permutations. BIT 24, 196–204 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Acad. Koptyug av. 4, Novosibirsk, 630090, Russia

    Elena Konstantinova & Alexey Medvedev

  2. Novosibirsk State University, Pirogova str. 2, Novosibirsk, 630090, Russia

    Elena Konstantinova

  3. Central European University, Nador ut. 9, Budapest, 1051, Hungary

    Alexey Medvedev

Authors
  1. Elena Konstantinova
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  2. Alexey Medvedev
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Correspondence to Alexey Medvedev.

Additional information

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). https://doi.org/10.1007/s00373-016-1679-x

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  • DOI: https://doi.org/10.1007/s00373-016-1679-x

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