Hamiltonian laceability with a set of faulty edges in hypercubes

Abstract: Faulty networks are useful because link or node faults can occur in a network. This paper examines the Hamiltonian properties of hypercubes under certain conditional faulty edges. Let consider the hypercube ( Q_n ), for ( n \geq 5 ) and set of faulty edges ( F ) such that ( |F| \leq 4n - 17 ). We prove that a Hamiltonian path exists connecting any two vertices in ( Q_n - F ) from distinct partite sets if they verify the next two conditions: (i) in $Q_n - F$ any vertex has a degree at least 2, and (ii) in $Q_n - F$ at most one vertex has a degree exactly equal to 2. These findings provide an understanding of fault-tolerant properties in hypercube networks.