A neighborhood and degree condition for pancyclicity and vertex pancyclicity

Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m, m + 1,..., n}, if G has a cycle of length l, then G is called [m, n]-pancyclic, and if G has a cycle of length l which contains u, then G is called [m, n]-vertex pancyclic. Let δ(G) be a minimum degree of G and let N_G(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81-91] Liu, Lou and Zhao proved that if |N_G(u) ∪ N_G(v)| + δ(G) ≥ n + 1 for any nonadjacent vertices u, v of G, then G is [3, n]-vertex pancyclic. In this paper, we prove if n ≥ 6 and |N_G(u)∪N_G(v)|+d_G(w) ≥ n for every triple independent vertices u, v, w of G, then (i) G is [3,n]-pancyclic or isomorphic to the complete bipartite graph K_n/2,n/2, and (ii) G is [5, n]-vertex pancyclic or isomorphic to the complete bipartite graph K_n/2,n/2.