Let G be a simple graph with n vertices. For any \documentclass[12pt]{minimal}
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\begin{document}$${v \in V(G)}$$\end{document} , let \documentclass[12pt]{minimal}
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\begin{document}$${N(v)=\{u \in V(G): uv \in E(G)\}}$$\end{document} , \documentclass[12pt]{minimal}
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\begin{document}$${NC(G)= \min \{|N(u) \cup N(v)|: u, v \in V(G)}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${uv \not \in E(G)\}}$$\end{document} , and \documentclass[12pt]{minimal}
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\begin{document}$${NC_2(G)= \min\{|N(u) \cup N(v)|: u, v \in V(G)}$$\end{document} and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any \documentclass[12pt]{minimal}
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\begin{document}$${u, v \in V(G)}$$\end{document} , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC2(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC2(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC2(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.
