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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 $n⩾2$ and $F$ is a subset of edges with $|F|. 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 $n⩾3$ and $F$ is a subset of edges with $|F|. 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.
##### MSC:
 68R10 Graph theory in connection with computer science (including graph drawing) 05C45 Eulerian and Hamiltonian graphs 68M15 Reliability, testing and fault tolerance computer systems
##### Keywords:
Hamiltonian laceable; Hypercube; Fault tolerance