Edge-disjoint paths in faulty augmented cubes

Abstract

Motivated by evaluating the reliability and fault tolerance of a network, we consider edge-disjoint paths in augmented cubes with faulty edges. We show that for any faulty edge set F⊂E(AQn) and δ(AQn−F)≥2, if |F|≤4n−8 for n≥4, there are min{degAQn−F(u),degAQn−F(v)} edge-disjoint paths connected any two vertices u and v in AQn−F, where degAQn−F(u) and degAQn−F(v) are the degree of vertices u and v in AQn−F, respectively. This result is optimal with respect to the maximum number of faulty edges.Simultaneously, we determine λ1(AQn) for n≥2, and λ2(AQn) for n≥4, where λg(AQn) is the g-extra edge-connectivity of AQn. Given a graph G and a non-negative integer g, the g-extra edge-connectivity of G, denoted λg(G), is the minimum cardinality of a set of edges in G, if it exists, whose deletion disconnects G and leaves each remaining component with more than g vertices.