Large vertex-flames in uncountable digraphs [article]

Florian Gut, Attila Joó
2022 arXiv   pre-print
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 Lovász: Let D=(V,E) be a finite digraph and let r∈ V. The local connectivity κ_D(r,v) from r to v is defined to be the maximal number of internally disjoint r→ v paths in D. A spanning subdigraph L of D with κ_L(r,v)=κ_D(r,v) for every v∈ V-r must have at least ∑_v∈ V-rκ_D(r,v) edges. Lovász proved that, maybe surprisingly, this lower bound is
more » ... sharp for every finite digraph. The optimality of an L sufficing the min-max criteria from Lovász' theorem may instead also be captured by the following structural characterization: For every v∈ V-r there is a system 𝒫_v of internally disjoint r→ v paths in L covering all the ingoing edges of v in L such that one can choose from each P∈𝒫_v either an edge or an internal vertex in such a way that the resulting set meets every r→ 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 ℵ_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.
arXiv:2107.12935v3 fatcat:rwhktnij5jex5kqdsxy5f44cgm