Abstract: n a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number of disjoint dijoins. However, prior to our work, it was not even known if at least 3 disjoint dijoins exist in an arbitrary digraph whose minimum dicut size is sufficiently large. By building connections with nowhere-zero (circular) k-flows, we prove that every digraph with minimum dicut size tau contains tau/k disjoint dijoins if the underlying undirected graph admits a nowhere-zero (circular) k-flow. The existence of nowhere-zero 6-flows in 2-edge-connected graphs (Seymour 1981) directly leads to the existence of tau/6 disjoint dijoins in any digraph with minimum dicut size tau, which can be found in polynomial time as well. The existence of nowhere-zero circular (2+1/p)-flows in 6p-edge-connected graphs (Lovász et al 2013) directly leads to the existence of tau*p/(2p+1) disjoint dijoins in any digraph with minimum dicut size tau whose underlying undirected graph is 6p-edge-connected.
This is joint work with Gérard Cornuéjols and R. Ravi, to appear at IPCO'24.