Gray codes avoiding matchings

Abstract : A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.
Type de document :
Article dans une revue
Discrete Mathematics and Theoretical Computer Science, DMTCS, 2009, 11 (2), pp.123--147
Liste complète des métadonnées

Littérature citée [14 références]  Voir  Masquer  Télécharger
Contributeur : Service Ist Inria Sophia Antipolis-Méditerranée / I3s <>
Soumis le : mercredi 7 mai 2014 - 16:17:20
Dernière modification le : mercredi 29 novembre 2017 - 10:26:17
Document(s) archivé(s) le : jeudi 7 août 2014 - 11:36:11


Fichiers produits par l'(les) auteur(s)


  • HAL Id : hal-00988206, version 1



Darko Dimitrov, Tomáš Dvořák, Petr Gregor, Riste Škrekovski. Gray codes avoiding matchings. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2009, 11 (2), pp.123--147. 〈hal-00988206〉



Consultations de la notice


Téléchargements de fichiers