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.