On hamiltonian-connected graphs

One of the most fundamental results concerning paths in graphs is due to Ore: In a graph G, if deg x + deg y ≧ |V(G)| + 1 for all pairs of nonadjacent vertices x, y ≅ V(G), then G is hamiltonian-connected. We generalize this result using set degrees. That is, for S ⊂ V(G), let deg S = |x≅SN(x)|, where N(x) = {v|xv ≅ E(G)} is the neighborhood of x. In particular we show: In a 3-connected graph G, if deg S1 + deg S2 ≧ |V(G)| + 1 for each pair of distinct 2-sets of vertices S1, S2 ⊂ V(G), then G is hamiltonian-connected. Several corollaries and related results are also discussed.