边端与图的拓扑端
Edge-ends versus topological ends of graphs
AI总结：
本文研究了无限图中拓扑端与边端的关系，通过组合方法刻画了拓扑端可嵌入边端空间的条件，并证明了拓扑端对应的边端特征。
摘要：
Diestel和Kühn证明了无限图的拓扑端正是其未被支配的图端，从而给出了拓扑端空间到图端空间的规范嵌入。对于由Hahn、Laviolette和Širáň引入的边端，这种嵌入一般不存在。本文中，我们刻画了那些拓扑端可以自然地映射到边端空间并兼容端空间之间规范映射的无限图的类别。我们的刻画是纯粹组合性的，并以顶点的边等价类来表达。此外，当这种嵌入存在时，我们精确地确定了哪些边端来自拓扑端，证明它们恰好是包含非支配射线的边端。这建立了与Diestel和Kühn定理在边端空间上的平行结果。
英文摘要：
Diestel and Kühn proved that the topological ends of an infinite graph are precisely its undominated graph ends, yielding a canonical embedding of the space of topological ends into the space of graph ends. For edge-ends, introduced by Hahn, Laviolette and Širáň, such an embedding does not exist in general. In this note, we characterize the class of infinite graphs for which the topological ends admit a natural injective map into the space of edge-ends that is compatible with the canonical maps between end spaces. Our characterization is purely combinatorial and is expressed in terms of edge-equivalence classes of vertices. Moreover, when such an embedding exists, we identify precisely which edge-ends arise from topological ends, showing that they are exactly the edge-ends containing a non-dominated ray. This establishes a parallel result to the theorem of Diestel and Kühn for edge-end spaces.
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