We show that a graph G has a normal spanning tree if and only if its vertex set is the union of countably many sets each separated from any subdivided infinite clique in G by a finite set of vertices. This proves a conjecture by Brochet and Diestel from 1994, giving a common strengthening of two classical normal spanning tree criterions due to Jung and Halin.

Moreover, our method gives a new, algorithmic proof of Halin's theorem that every connected graph not containing a subdivision of a countable clique has a normal spanning tree.

中文翻译：

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普通生成树的统一存在定理

我们证明，当且仅当图G的顶点集是无数个集合的并集时，该图才具有正常的生成树。每个集合与G中细分的无限集团之间由一个有限的顶点集隔开。这证明了Brochet和Diestel从1994年开始的猜想，归因于Jung和Halin共同加强了两个经典的正常生成树标准。

此外，我们的方法为Halin定理提供了一种新的算法证明，即每个不包含可数集团细分的连接图都具有正常的生成树。