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Fault-tolerant Hamiltonian laceability of hypercubes. (English) Zbl 1043.68081
Summary: It is known that every hypercube Q n is a bipartite graph. Assume that n2 and F is a subset of edges with |F|<n-2. We prove that there exists a Hamiltonian path in Q n -F between any two vertices of different partite sets. Moreover, there exists a path of length 2 n -2 between any two vertices of the same partite set. Assume that n3 and F is a subset of edges with |F|<n-3. We prove that there exists a Hamiltonian path in Q n -v-F between any two vertices in the partite set without v· Furthermore, all bounds are tight.
68R10Graph theory in connection with computer science (including graph drawing)
05C45Eulerian and Hamiltonian graphs
68M15Reliability, testing and fault tolerance computer systems