Hamiltonian paths and hamiltonian connectivity in graphs (Q1309475)

A degree and neighborhood type condition on independent triples of vertices of a graph \(G\) that implies \(G\) is hamiltonian-connected is given. In particular, the following is proved. If \(G\) is a 3-connected graph of order \(n\) such that \(d(u) + d(v) + d(w)-| N(u) \cap N(v) \cap N(w) | \geq n+1,\) then \(G\) is hamiltonian-connected. This generalizes the classical degree condition result of Ore as well as other later extensions of this result.