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 P _i[u, v] denote the path with i vertices which connects u and v. In this paper, we prove that if n ≥ 5 and |N _G(u)∪N _G(v)| +d _G(w) ≥ n+1 for every triple of independent vertices u,v,w of G, then there exists a P _i[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].