Strong Menger connectivity with conditional faults of folded hypercubes

Motivated by parallel routing in networks with faults and evaluating the reliability of networks, we consider strong Menger connectivity of the folded hypercube networks. We show that in all n-dimensional folded hypercubes with a vertex set S of n−1 vertices removed, each pair of unremoved vertices x and y are connected by min⁡{dG−S(x),dG−S(y)} vertex-disjoint paths (i.e., strong Menger property), where dG−S(x) and dG−S(y) are the remaining degree of vertices x and y in G−S, respectively. Moreover, if there are 2n−3 vertex faults, and each vertex except for the vertex faults has at least two fault-free adjacent vertices, then all folded hypercube networks still have the strong Menger property.