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Enlarging vertex-flames in countable digraphs
Journal of Combinatorial Theory Series B ( IF 1.4 ) Pub Date : 2021-07-13 , DOI: 10.1016/j.jctb.2021.06.011
Joshua Erde , J. Pascal Gollin , Attila Joó

A rooted digraph is a vertex-flame if for every vertex v there is a set of internally disjoint directed paths from the root to v whose set of terminal edges covers all ingoing edges of v. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. Calvillo-Vives rediscovered and extended this theorem proving that every vertex-flame of a given finite rooted digraph can be extended to be large. The analogue of Lovász' result for countable digraphs was shown by the third author where the notion of largeness is interpreted in a structural way as in the infinite version of Menger's theorem. We give a common generalisation of this and Calvillo-Vives' result by showing that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame.



中文翻译:

在可数有向图中扩大顶点火焰

一个有根有向图是一个顶点火焰,如果对于每个顶点v都有一组从根到v的内部不相交的有向路径,它们的一组终端边覆盖了v 的所有进入边. Lovász 表明,每个有限有根有向图都允许一个跨越子有向图,它是一个顶点火焰和大的,其中后者意味着它保留了从根到每个顶点的局部连接。Calvillo-Vives 重新发现并扩展了这个定理,证明给定的有限有根有向图的每个顶点火焰都可以扩展到很大。第三作者展示了 Lovász 的可数有向图结果的类似物,其中大的概念以结构方式解释,就像在门格尔定理的无限版本中一样。我们通过证明在每个可数有根有向图中每个顶点火焰都可以扩展到一个大顶点火焰,给出了这个和 Calvillo-Vives 结果的共同推广。

更新日期:2021-07-13
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