Some panconnected and pancyclic properties of graphs with a local ore-type condition

Asratian and Khachatrian proved that a connected graph G of order at least 3 is hamiltonian if d(u) + d(v);;::: JN(u) U N(v) U N.(w)I for any path uwv with uv ff E(G), .where N(x) is the neighborhood of a vertex x. We prove that a graph G with this condition, which is not complete bipartite, has the following properties: a) For each pair of vertices x, y with distance d(x, y) 2:: 3 and for each integer n, d(x, y) :;;; n :;;; I V(G)I 1, there is an x y path of length n. (b) For each edge e which does not lie on a triangle and for each n, 4 :;;; n :;;; I V(G)I, there is a cycle of length n containing e. (c) Each vertex oft;; lies on a cycle of every length from 4 to I V(G)j. This implies that G is vertex pancyclic if and only if each vertex of G lies on a triangle.