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

Abstract

AbstractBerge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection Ck 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.