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Colouring of cycles in the de Bruijn graphs

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We show that the problem of finding the family of all so called the locally reducible factors in the binary de Bruijn graph of order k is equivalent to the problem of finding all colourings of edges in the binary de Bruijn graph of order k-1, where each vertex belongs to exactly two cycles of different colours. In this paper we define and study such colouring for the greater class of the de Bruijn graphs in order to define a class of so called regular factors, which is not so difficult to construct. Next we prove that each locally reducible factor of the binary de Bruijn graph is a subgraph of a certain regular factor in the m-ary de Bruijn graph.
Twórcy
autor
  • Department of Applied Mathematics, Technical University of Lublin, Bernardyńska 13, 20-950 Lublin, Poland
  • Department of Applied Mathematics, Technical University of Lublin, Bernardyńska 13, 20-950 Lublin, Poland
Bibliografia
  • [1] M. Cohn and A. Lempel, Cycle decomposition by disjoint transpositions, J. Combin. Theory (A) 13 (1972) 83-89, doi: 10.1016/0097-3165(72)90010-6.
  • [2] E.D. Erdmann, Complexity measures for testing binary keystreams, PhD thesis, Stanford University, 1993.
  • [3] H. Fredricksen, A survey of full length nonlinear shift register cycle algorithms, SIAM Rev. 24 (1982) 195-221, doi: 10.1137/1024041.
  • [4] E.R. Hauge and T. Helleseth, De Bruijn sequences, irreducible codes and cyclotomy, Discrete Math. 159 (1996) 143-154, doi: 10.1016/0012-365X(96)00106-9.
  • [5] C.J.A. Jansen, Investigations on nonlinear strimcipher systems: Construction and evaluation methods, PhD thesis, Technical University of Delft, 1989.
  • [6] M. Łatko, Design of the maximal factors in de Bruijn graphs, (in Polish), PhD thesis, UMCS, 1987.
  • [7] E. Łazuka and J. Żurawiecki, The lower bounds of a feedback function, Demonstratio Math. 29 (1996) 191-203.
  • [8] R.A. Rueppel, Analysis and design of stream ciphers (Springer-Verlag, 1986).
  • [9] P. Wlaź and J. Żurawiecki, An algorithm for generating M-sequences using universal circuit matrix, Ars Combinatoria 41 (1995) 203-216.
  • [10] J. Żurawiecki, Elementary k-iterative systems (the binary case), J. Inf. Process. Cybern. EIK 24 1/2 (1988) 51-64.
  • [11] J. Żurawiecki, Locally reducible iterative systems, Demonstratio Math. 23 (1990) 961-983.
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1103
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