Degree with Neighborhood Conditions and Highly Hamiltonian Graphs

In 1997 Bollobas and Thomason (J. Graph Theory 26: 165-173, 1997) and Brandt (Discrete Appl. Math. 79: 63-66, 1997) defined the weakly pancyclic. In this paper we define weakly vertex-pancyclic and obtain a new sufficient condition for graph to be weakly vertex-pancyclic as the following: if G is a 2-connected graph of order n, and {vertical bar N(u) boolean OR N(v)vertical bar + d(w) : u, v, w is an element of V (G), uv is not an element of E(G), wu, or wv is not an element of E(G)} >= n + 1, then G is weakly vertex-pancyclic. This result also implies a conjecture of Faudree et al.