A Note of Reliability Analysis of SM-λ in Folded-Crossed Hypercube with Conditional Faults

Abstract

The fault tolerance of an interconnection network of parallel and distributed systems can be evaluated by various topological parameters of its underlying graph [Formula: see text], with strong Menger edge connectivity being a vital parameter in this regard. A connected graph [Formula: see text] is called strongly Menger edge connected (SM-[Formula: see text]) if it connects any pair of vertices [Formula: see text] and [Formula: see text] with [Formula: see text] number of edge-disjoint paths. Under the uniform distribution of faults in a large interconnection network, it is improbable that each faulty edge incident to a vertex will occur simultaneously. Thus, [Formula: see text]-strongly Menger edge connected of order [Formula: see text] was introduced in 2018 by He et al. Here, [Formula: see text] is called as [Formula: see text]-strongly Menger edge connected of order [Formula: see text], if [Formula: see text] remains SM-[Formula: see text], where [Formula: see text] is an arbitrary edge set in a graph [Formula: see text] with [Formula: see text] and the minimum degree of the remaining graph [Formula: see text]. The largest [Formula: see text] keeping the property of [Formula: see text] being SM-[Formula: see text] is denoted as [Formula: see text]. Among variants of hypercube, the [Formula: see text]-dimensional folded-crossed hypercube [Formula: see text] attracts attention in recent years. In this paper, we focus on calculating the exact value of the maximum conditional edge-fault-tolerant number of order [Formula: see text] of [Formula: see text], [Formula: see text] for two integers [Formula: see text] and [Formula: see text].