On the Infinite Lucchesi-Younger Conjecture I

TL;DR

This paper explores a conjecture extending the Lucchesi-Younger theorem to infinite directed graphs, proposing a structural approach and proving special cases, advancing understanding of dicut properties in infinite digraphs.

Contribution

It introduces a conjecture for infinite digraphs based on a structural view and reduces the problem to countable, 2-connected underlying graphs, with partial proofs.

Findings

Conjecture reduces to countable, 2-connected digraphs.

Proved several special cases of the conjecture.

Provides a structural perspective on infinite dicut min-max relations.

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.