Disjoint dijoins

TL;DR

This paper explores a conjecture about disjoint dijoins in directed graphs, proving it in special cases and discussing related counterexamples and conditions.

Contribution

It introduces a conjecture linking connectivity of a subset of edges to disjoint dijoins and proves it for planar graphs and caterpillar subdivisions.

Findings

Proved the conjecture for planar graphs.

Proved the conjecture for subdivisions of caterpillars.

Discussed counterexamples and the importance of connectivity.

Abstract

A dijoin in a digraph is a set of edges meeting every directed cut. D. R. Woodall conjectured in 1976 that if G is a digraph, and every directed cut of G has at least k edges, then there are k pairwise disjoint dijoins. This remains open, but a capacitated version is known to be false. In particular, A. Schrijver gave a digraph G and a subset S of its edge-set, such that every directed cut contains at least two edges in S, and yet there do not exist two disjoint dijoins included in S. In Schrijver's example, G is planar, and the subdigraph formed by the edges in S consists of three disjoint paths. We conjecture that when k = 2, the disconnectedness of S is crucial: more precisely, that if G is a digraph, and S is a subset of the edges of G that forms a connected subdigraph (as an undirected graph), and every directed cut of G contains at least two edges in S, then we can partition S into two dijoins. We prove this in two special cases: when G is planar, and when the subdigraph formed by the edges in S is a subdivision of a caterpillar.