Node-to-node disjoint paths in hypercube networks with faulty edges

Let u be the smallest degree node in a hypercube network with some faulty edges. Suppose that the number of fault-free edges incident with u is not more than that of the others, and denote this number by k. In this article, we mainly concentrate on the conditional hypercube and folded hypercube. In view of this purpose, we prove that there exist k disjoint paths between the smallest degree node and its neighbour which contain all nodes in an n -cube with at most n-2 faulty edges. Based on the result, it has been found that for the smallest degree node and its neighbour there are k internal-disjoint oaths between them, which traverse all vertices of an n-dimensional folded cube with at most n-1 faulty edges.