A proof for Berges Dual Conjecture for Bipartite Digraphs

Given a (vertex)-coloring C = C 1 , C 2 , ...C m of P a digraph D and a positive integer k, the k-norm of C is defined as C k = m i=1 min C i, k. A coloring C is k-optimal if its k-norm C k is minimum over all colorings. A (path) k-pack P k is a collection of at most k vertex-disjoint paths. A coloring C and a k-pack P k are orthogonal if each color class intersects as many paths as possible in P k , that is, if C i k, C i P j = 1 for every path P j P k , otherwise each vertex of C i lies in a different path of P k . In 1982, Berge conjectured that for every k-optimal coloring C there is a k-pack P k 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.