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 ofG and let NG(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 |NG(u) ∪ NG(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 |NG(u)∪NG(v)|+dG(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 Kn/2,n/2, and (ii) G is [5, n]-vertex pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2.