A Unified Approach to Known and Unknown Cases of Berge’s Conjecture Eli Berger * and Irith Ben-Arroyo Hartman † Abstract Berge’s elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing ∑ 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 λ is 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 [5] for k = 2. There was no proof which covers all the known cases of Berge’s conjecture. In this paper, 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. 1 Introduction Dilworth’s well known theorem [8] states that in a partially ordered set the size of a maximum antichain equals the size of a minimum chain partition. Greene and Kleitman [13] generalized Dilworth’s Theorem to a set of k antichains, where k is a fixed positive integer. Linial [15] conjectured that the Greene-Kleitman Theorem could be extended to all digraphs by replacing the equality by an inequality. Berge [4] made a stronger conjecture than Linial’s. (For precise statements of all the mentioned theorems, see Section 2). Berge’s conjecture was proved for all acyclic digraphs; see [15], [7],[17], [6] and [2]. For k = 1, Berge’s Conjecture holds by the Gallai-Milgram Theorem [11]. Let λ be the number of vertices in the longest dipath in a digraph. When k ≥ λ, Berge’s Conjecture holds by the Gallai-Roy theorem [10, 16]. Recently, the authors [5] proved Berge’s conjecture for k = 2. Linial’s conjecture for k = λ − 1 was also proved by Seb˝ o and the second author, using a powerful theorem of Seb˝ o [19]. For other values of k, Berge’s Conjecture and Linial’s Conjecture are open. For a survey of the subject see [14]. Since the Gallai-Milgram and the Gallai-Roy theorems are special cases of Berge’s conjecture, then any algorithmic proof for Berge’s conjecture must include, as special cases, algorithmic proofs for * Department of Mathematics, Faculty of Science, University of Haifa, Haifa 31905, Israel. e-mail: berger@math.haifa.ac.il † Caesarea Rothschild Institute for Interdisciplinary Applications of Computer Science, University of Haifa, Haifa 31905, Israel, and Department of CSA, Indian Institute of Science, Bangalore, India. e-mail: irith.hartman@gmail.com 1