Disjoint paths in the enhanced hypercube with a faulty subgraph

Problems about embedding of disjoint paths in interconnection networks have received much attention in recent years. A connected graph G is strong Menger connected if there are min{d(G)(u), d(G)(v)} internally disjoint paths joining any two distinct vertices u and v in G. The enhanced hypercube Q(n, k) is an important variant of the hypercube Q(n) that retains many desirable properties of the hypercube. In order to study its fault tolerance, we consider the problem of embedding internally disjoint paths in an enhanced hypercube when part of the network is faulty. We show that the subgraph obtained from the enhanced hypercube Q(n, k) (2 <= k <= n) by deleting the vertices of a faulty subnetwork Q(s) (1 <= s <= n - 1) or Q(s, k) (k <= s <= n - 1) is strong Menger connected.