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Given a (vertex)-coloring C = {C1, C2, ...Cm} of a digraph D and a positive integer k, the k-norm of C is defined as |C|k = ∑m i=1 min{|Ci|, k}. A coloring C is k-optimal if its k-norm |C|k is minimum over all colorings. A (path) k-pack P is a collection of at most k vertex-disjoint paths. A coloring C and a k-pack P are orthogonal if each color class intersects as many paths as possible in P, that is, if |Ci| ≥ k, |Ci ∩ Pj| = 1 for every path Pj ∈ P, otherwise each vertex of Ci lies in a different path of P . In 1982, Berge conjectured that for every k-optimal coloring C there is a k-pack P orthogonal to C. This conjecture is false for arbitrary digraphs, having a counterexample with odd cycle. In this paper we prove this conjecture for bipartite digraphs.

A proof for Berges Dual Conjecture for Bipartite Digraphs