Some localization theorems on hamiltonian circuits

Theorems on the localization of the conditions of G. A. Dirac ( Proc. London Math. Soc. (3), 2 1952, 69–81), O. Ore ( Amer. Math. Monthly, 67 1960, 55), and Geng-hua Fan ( J. Combin. Theory Ser. B, 37 1984, 221–227) for a graph to be hamiltonian are obtained. It is proved, in particular, that a connected graph G on p ≥ 3 vertices is hamiltonian if d(u) ≥ ∥M 3(u)∥ 2 for each vertex u in G, where M 3( u) is the set of vertices v in G that are a distance at most three from u.