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 $G$, with strong Menger edge connectivity being a vital parameter in this regard. A connected graph $G$ is called strongly Menger edge connected (SM-$\lambda$) if it connects any pair of vertices $u$ and $v$ with $min\{d_G(u),d_G(v)\}$ 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, $m$-strongly Menger edge connected of order $t$ was introduced in 2018 by He et al. Here, $G$ is called as $m$-strongly Menger edge connected of order $t$, if $G-F$ remains SM-$\lambda$, where $F$ is an arbitrary edge set in a graph $G$ with $\left|F\right|\le m$ and the minimum degree of the remaining graph $\delta (G-F)\ge t$. The largest $m$ keeping the property of $G-F$ being SM-$\lambda$ is denoted as $s{m}_{\lambda }^t(G)$. Among variants of hypercube, the $n$-dimensional folded-crossed hypercube $FC{Q}_n$ 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 $t$ of $FC{Q}_n$, $s{m}_{\lambda }^t(FC{Q}_n)=2^t(n+1-t)-(n+1)$ for two integers $1\le t\le n-2$ and $n\ge 3$.