Embedded paths and cycles in faulty hypercubes

Nelson Castañeda () and Ivan S. Gotchev ()
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Nelson Castañeda: Central Connecticut State University
Ivan S. Gotchev: Central Connecticut State University

Journal of Combinatorial Optimization, 2010, vol. 20, issue 3, No 2, 224-248

Abstract: Abstract An important task in the theory of hypercubes is to establish the maximum integer f n such that for every set ℱ of f vertices in the hypercube ${\mathcal {Q}}_{n},$ with 0≤f≤f n , there exists a cycle of length at least 2 n −2f in the complement of ℱ. Until recently, exact values of f n were known only for n≤4, and the best lower bound available for f n with n≥5 was 2n−4. We prove that f 5=8 and obtain the lower bound f n ≥3n−7 for all n≥5. Our results and an example provided in the paper support the conjecture that $f_{n}={n\choose 2}-2$ for each n≥4. New results regarding the existence of longest fault-free paths with prescribed ends are also proved.

Keywords: Hypercube; Fault tolerant embedding; Maximal paths with prescribed ends (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s10878-008-9205-6

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