可数图中的Lovász-Cherkassky定理
The Lovász-Cherkassky theorem in countable graphs
摘要：
Lovász和Cherkassky在20世纪70年代独立地发现，如果$ G $是一个有限图，具有给定的终端顶点集合$ T $，使得$ G $是内欧拉的，那么连接$ T $中不同顶点的边不相交路径的最大数量是$ \sum_{t\in T}λ(t, T-t) $，其中$λ$是局部边连通度函数。在Lovász-Cherkassky定理中，边不相交的$ T $ -路系统的最优性可以通过门格尔定理证明某些截集的存在性。Aharoni和Berger对门格尔定理的无限推广（早期被称为Erdens-Menger猜想）以及Nash-Williams对无限欧拉图的刻画使得有可能以结构化的方式将该定理推广到无限图。本文的目的是制定这一推广，并证明它的可数图。
英文摘要：
Lovász and Cherkassky discovered in the 1970s independently that if $ G $ is a finite graph with a given set $ T $ of terminal vertices such that $ G $ is inner Eulerian, then the maximal number of edge-disjoint paths connecting distinct vertices in $ T $ is $ \sum_{t\in T}λ(t, T-t) $ where $λ$ is the local edge-connectivity function. The optimality of a system of edge-disjoint $ T $-paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Conjecture) together with the characterization of infinite Eulerian graphs due to Nash-Williams makes it possible to generalise the theorem for infinite graphs in a structural way. The aim of this paper is to formulate this generalisation and prove it for countable graphs.
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