Large vertex-flames in uncountable digraphs

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 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_D (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 sharp for every finite digraph.

The optimality of an L satisfying 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 ${{\cal P}_v}$ of internally disjoint r → v paths in L covering all the ingoing edges of v in L such that one can choose from each $P \in {{\cal P}_v}$ 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 ${\aleph _1}$ 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 away.