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Long cycles in hypercubes with optimal number of faulty vertices

Author

Listed:
  • Jiří Fink

    (Charles University in Prague)

  • Petr Gregor

    (Charles University in Prague)

Abstract

Let f(n) be the maximum integer such that for every set F of at most f(n) vertices of the hypercube Q n , there exists a cycle of length at least 2 n −2|F| in Q n −F. Castañeda and Gotchev conjectured that $f(n)=\binom{n}{2}-2$ . We prove this conjecture. We also prove that for every set F of at most (n 2+n−4)/4 vertices of Q n , there exists a path of length at least 2 n −2|F|−2 in Q n −F between any two vertices such that each of them has at most 3 neighbors in F. We introduce a new technique of potentials which could be of independent interest.

Suggested Citation

  • Jiří Fink & Petr Gregor, 2012. "Long cycles in hypercubes with optimal number of faulty vertices," Journal of Combinatorial Optimization, Springer, vol. 24(3), pages 240-265, October.
    • Handle: RePEc:spr:jcomop:v:24:y:2012:i:3:d:10.1007_s10878-011-9379-1
    DOI: 10.1007/s10878-011-9379-1
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