On spanning connected graphs

A k-container  C(u,v)C(u,v) of G   between uu and vv is a set of k   internally disjoint paths between uu and vv. A k  -container C(u,v)C(u,v) of G   is a k*k*-container   if the set of the vertices of all the paths in C(u,v)C(u,v) contains all the vertices of G. A graph G   is k*k*-connected   if there exists a k*k*-container between any two distinct vertices. Therefore, a graph is 1*1*-connected (respectively, 2*2*-connected) if and only if it is hamiltonian connected (respectively, hamiltonian). In this paper, a classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian (respectively, hamiltonian connected), is generalized to k*k*-connected graphs.