On graphs satisfying a local ore-type condition

For an integer i, a graph is called an Li-graph if, for each triple of vertices u, v, w with d(u, v) = 2 and w ∈ N(u) ∩N(v), d(u) + d(v) ≥ |N(u) ∪N(v) ∪N(w)| − i. Asratian and Khachatrian proved that connected L0-graphs of order at least 3 are hamiltonian, thus improving Ore’s Theorem. All K1,3-free graphs are L1-graphs, whence recognizing hamiltonian L1-graphs is an NP-complete problem. The following results about L1-graphs, unifying known results of Ore-type and known results on K1,3-free graphs, are obtained. Set K = {G | Kp,p+1 ⊆ G ⊆ Kp ∨ Kp+1 for some p ≥ 2 } (∨ denotes join ). If G is a 2-connected L1-graph, then G is 1-tough unless G ∈ K. Furthermore, if G is a connected L1-graph of order at least 3 such that |N(u) ∩ N(v)| ≥ 2 for every pair of vertices u, v with d(u, v) = 2, then G is hamiltonian unless G ∈ K, and every pair of vertices x, y with d(x, y) ≥ 3 is connected by a Hamilton path. This result implies that of Asratian and Khachatrian. Finally, if G is a connected L1-graph of even order, then G has a perfect matching.