LOCALIZATION THEOREM ON HAMILTONIAN GRAPHS

Let G be a 2-connected graph of order n(≥3). If I(u,v) ≥S(u,v) or max {d(u), d(v)} ≥n/2 for any two vertices u, v at distance two in an induced subgraph H1,3 or P3 of G, then G is hamiltonian. Here I(u, v) = |N(u) n N(v)|, S(n, v) denotes the number of edges of mtximum star containing u, v as an induced subgraph in G.