Hamiltonian laceability with a set of faulty edges in hypercubes

TL;DR

This paper investigates the Hamiltonian connectivity of hypercube networks with a limited number of faulty edges, establishing conditions under which Hamiltonian paths still exist despite faults.

Contribution

It proves that hypercubes retain Hamiltonian paths between certain vertices under specific fault conditions, extending fault-tolerance understanding in hypercube networks.

Findings

Hamiltonian paths exist under certain fault conditions in hypercubes.

Fault-tolerance is maintained if each vertex has degree ≥ 2 and at most one vertex has degree exactly 2.

Results apply to hypercubes with up to 4n - 17 faulty edges for n ≥ 5.

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.