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Gray Codes and Symmetric Chains

Authors Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, Kaja Wille

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Petr Gregor
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic
Sven Jäger
  • Institut für Mathematik, Technische Universität Berlin, Germany
Torsten Mütze
  • Institut für Mathematik, Technische Universität Berlin, Germany
Joe Sawada
  • School of Computer Science, University of Guelph, Canada
Kaja Wille
  • Institut für Mathematik, Technische Universität Berlin, Germany

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Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, and Kaja Wille. Gray Codes and Symmetric Chains. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 66:1-66:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1-l,n+l], where 1 <= l <= n+1, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (l=1). We provide a solution for the case l=2 and solve a relaxed version of the problem for general values of l, by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the n-dimensional hypercube for any n >= 12.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Gray code
  • Hamilton cycle
  • hypercube
  • poset
  • symmetric chain


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