On the infinite Lucchesi–Younger conjecture I

Abstract

A dicut in a directed graph is a cut for which all of its edges are directed to a common side of the cut. A famous theorem of Lucchesi and Younger states that in every finite digraph the least size of an edge set meeting every dicut equals the maximum number of disjoint dicuts in that digraph. In this first paper out of a series of two papers, we conjecture a version of this theorem using a more structural description of this min-max property for finite dicuts in infinite digraphs.We show that this conjecture can be reduced to countable digraphs where the underlying undirected graph is 2-connected, and we prove several special cases of the conjecture.

Disjoint dijoins for classes of dicuts in finite and infinite digraphs

“…Given a capacity c of D, we call the restriction of c to E(D/≡ B ) the capacity of D/≡ B induced by c. We shall use the following observation from [GH21] about this digraph. Proposition 6.2.…”

“…Instead of focusing on specific classes of digraphs, one other possible avenue to explore Conjecture 1 is to restrict the attention not to all dicuts, but to some specific classes of dicuts of a digraph. In [GH21,GH22], Gollin and Heuer considered a similar approach regarding classes of dicuts in their attempt of generalising the theorem of Lucchesi and Younger to infinite digraphs.…”

Disjoint dijoins for classes of dicuts in finite and infinite digraphs

“…Note that D{" B does not contain any directed cycles. Given a capacity c of D, we call the restriction of c to EpD{" B q the capacity of D{" B induced by c. We shall use the following observation from [10] about this digraph. The main tool we use to extend results about Question 2 from classes of finite graphs to finitary infinite versions is the following compactness-type lemma.…”

“…Instead of focusing on specific classes of digraphs, one other possible avenue to explore Conjecture 1 is to restrict the attention not to all dicuts, but to some specific classes of dicuts of a digraph. In [10,11], the first two authors considered a similar approach regarding classes of dicuts in their attempt of generalising the theorem of Lucchesi and Younger to infinite digraphs.…”