Hamiltonian laceability with a set of faulty edges in hypercubes

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 $n\ge 5$ and set of faulty edges $F$ such that $\mathrm{\mid }F\mathrm{\mid }\le 4n-17$. 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.