Abstract Let G=(V, E) be a graph and u, v be two arbitrary vertices of V (G). Then G is hamilton-connected if there exists a spanning path between u and v, and G is hamiltonian if there exist two internally disjoint pathes between u and v and the union of these two paths spans V (G). More generally, G is said to be spanning k-connected if there exist k internally disjoint pathes between u and v and the union of these k pathes contains all vertices of G. In the paper, we first generalize a classic theorem of Vergnas on hamiltonian graphs to spanning k-connectedness. Furthermore, we determine extremal number of edges in a spanning k-connected graph by extending an old theorem due to Erdős. Finally, we partially establish spanning k-connected versions of famous Chvátal-Erdős theorem.