## 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⊆ E(G) 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.

Original language | English (US) |
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Pages (from-to) | 18-35 |

Number of pages | 18 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 120 |

DOIs | |

State | Published - Sep 1 2016 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

## Keywords

- Disjoint dijoins
- Feedback arc set
- Lucchesi-Younger theorem
- Woodall's conjecture