Distance between two k-sets and path-systems extendibility

Given a simple graph G on n vertices, let sigma(2)(G) be the minimum sum of the degrees of any two non adjacent vertices. The graph C is said to be connected if any two distinct vertices may be joined by a path. It is easy to see that if sigma(2)(G) >= n - 1 then G is not only connected, but we can choose the connecting path to be of size at most two. Ore [4] proved that if sigma 2(G) >= n + 1 we may always choose this path to cover all the vertices of G. In this paper we extend these results to systems of vertex disjoint paths connecting two vertex k-sets of G.