A neighborhood and degree condition for panconnectivity

Let G be a 2-connected graph of order n with x, y ∈ V (G). For u, v ∈ V (G), let Pi[u, v] denote the path with i vertices which connects u and v. In this paper, we prove that if n ≥ 5 and |NG(u)∪NG(v)|+dG(w) ≥ n+1 for every triple of independent vertices u, v, w of G, then there exists a Pi[x, y] in G for 5 ≤ i ≤ n, or G belongs to one of three exceptional classes. This implies a positive answer to a conjecture by Wei and Zhu [Graphs Combin. 14 (1998), 263–274].