Independent Even Cycles in the Pancake Graph and Greedy Prefix-Reversal Gray Codes
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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.
KeywordsPancake graph Pancake sorting Even cycles Disjoint cycle cover Prefix-reversal Gray codes Fastening cycle
Mathematics Subject Classification05C45 05C25 05C38 90B10
- 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)Google Scholar
- 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)Google Scholar
- 9.Skiena, S.: Hypercubes, Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, pp. 148–150. Addison–Wesley, Reading (1990)Google Scholar