A unified approach to known and unknown cases of Berge's conjecture

Berge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing Σ_P∈P min{|P|,k}, there exists a collection C^k of k disjoint independent sets, where each dipath P∈P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linial's conjecture, the Greene-Kleitman Theorem and Dilworth's Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the Gallai-Milgram Theorem, for k≥λ (where λis the number of vertices in the longest dipath in the graph), by the Gallai-Roy Theorem, and when the optimal path partition P contains only dipaths P with |P|≥k. Recently, it was proved (Eur J Combin (2007)) for k=2. There was no proof that covers all the known cases of Berge's conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=λ-1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.