Some new localization theorems on Hamiltonian properties

Let G be a 2-connected graph with n ?: 3 vertices such that for any two vertices u, v at distance two in an induced subgraph K 1,3 or P4 of G, the inequality d(u) + d(v) ?: IN(u) U N(v) U N(w)ls holds for all w E N(u) n N(v). We prove that (i) if s = 1 and IN(u) n N(v)1 ?: 2, then G is hamiltonian or K p,p+1 ~ G ~ Kp + K p+1; (ii) if s = 0, then G is either pancyclic, or bipartite graph. This generalizes two localization theorems known before.