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Given a (vertex)-coloring $\mathcal{C} = \{C_{1}, C_{2}, ... C_{m}\}$ of a digraph $D$ and a positive integer $k$, the $k$-norm of $\mathcal{C}$ is defined as  $ |\mathcal{C}|_k = \sum_{i = 1}^{m} min\{|C_i|, k\}.$ A coloring $\mathcal{C}$ is $k$-optimal if its $k$-norm  $|\mathcal{C}|_k$ is minimum over all colorings. A (path) $k$-pack  $\mathcal{P}^k$ is a collection of at most $k$ vertex-disjoint paths. A coloring $\mathcal{C}$ and a $k$-pack $\mathcal{P}^k$ are orthogonal if each color class intersects as many paths as possible in $\mathcal{P}^k$, that is, if $|C_i| \ge k$, $|C_i \cap P_j| = 1$ for every path $P_j \in \mathcal{P}^k$, otherwise each vertex of $C_i$ lies in a different path of $\mathcal{P}^k$. In 1982, Berge conjectured that for every $k$-optimal coloring $\mathcal{C}$ there is a $k$-pack $\mathcal{P}^k$ orthogonal to $\mathcal{C}$. This conjecture is false for arbitrary digraphs, having a counterexample with odd cycle. In this paper we prove this conjecture for bipartite digraphs. In addition we show that the conjecture cannot hold for perfect graphs by exhibiting a counterexample.

In 1982, Berge defined the class of χ-diperfect digraphs. A digraph D is χ-diperfect if for every induced subdigraph H of D and every minimum coloring S of H there exists a path P of H with exactly one vertex of each color class of S. Berge also showed examples of non-χ-diperfect orientations of odd cycles and their complements. The ultimate goal in this research area is to obtain a characterization of χ-diperfect digraphs in terms of forbidden induced subdigraphs. In this work, we give steps towards this goal by presenting characterizations of orientations of odd cycles and their complements that are χ-diperfect. We also show that certain classes of digraphs are χ-diperfect. Moreover, we present minimal non-χ-diperfect digraphs that were unknown.