Large vertex-flames in uncountable digraphs

TL;DR

This paper extends the concept of large vertex-flames from finite and countably infinite digraphs to uncountable digraphs of size 1, using advanced techniques to address the increased complexity.

Contribution

It provides the first structural characterization of large vertex-flames in uncountable digraphs, advancing the understanding from countable to uncountable cases.

Findings

Extended Love1sz's theorem to 1-sized digraphs

Developed new techniques for uncountable graph structures

Achieved structural characterization for uncountable large flames

Abstract

The study of minimal subgraphs witnessing a connectivity property is an important field in graph theory. The foundation for large flames has been laid by Lov\'asz: Let $ D=(V,E) $ be a finite digraph and let $ r\in V $. The local connectivity $ \kappa_D(r,v) $ from $ r $ to $ v $ is defined to be the maximal number of internally disjoint $ r\rightarrow v $ paths in $ D $. A spanning subdigraph $ L $ of $ D $ with $ \kappa_L(r,v)=\kappa_D(r,v) $ for every $ v\in V-r $ must have at least $\sum_{v\in V-r}\kappa_D(r,v) $ edges. Lov\'asz proved that, maybe surprisingly, this lower bound is sharp for every finite digraph. The optimality of an $ L $ sufficing the min-max criteria from Lov\'asz' theorem may instead also be captured by the following structural characterization: For every $ v\in V-r $ there is a system $ \mathcal{P}_v $ of internally disjoint $ r\rightarrow v $ paths in $ L $ covering all the ingoing edges of $ v $ in $ L $ such that one can choose from each $ P\in \mathcal{P}_v $ either an edge or an internal vertex in such a way that the resulting set meets every $ r\rightarrow v $ path of $ D $. The positive result for countably infinite digraphs based on this structural infinite generalisation were obtained by the second author. In this paper we extend this to digraphs of size $ \aleph_1 $ which requires significantly more complex techniques. Despite solving yet the smallest uncountable case, the complete understanding of the concept and potentially a proof for arbitrary cardinality still seems to be far.