Embedded paths and cycles in faulty hypercubes

An important task in the theory of hypercubes is to establish the maximum integer fn such that for every set F of f vertices in the hypercube Qn, with 0 ≤ f ≤ fn, there exists a cycle of length at least 2n − 2f in the complement of F. Until recently, exact values of fn were known only for n ≤ 4, and the best lower bound available for fn with n ≥ 5 was 2n−4. We prove that f5 = 8 and obtain the lower bound fn ≥ 3n − 7 for all n ≥ 5. Our results and an example provided in the paper support the conjecture that fn = n 2 − 2 for each n ≥ 4. New results regarding the existence of longest fault-free paths with prescribed ends are also proved.

• Type: Journal
• Year: 2010
• Where: JCO
• Authors: Nelson Castañeda, Ivan S. Gotchev
• Keywords: JCO 2010, Lower Bound, Lower Bound Fn, Maximum Integer Fn