Let $D$ be a digraph. A collection of disjoint sets of vertices (respec., collection of disjoint subdigraphs) $\mathcal{H}$ of $D$ and a vertex subset (or subdigraph) $Q$ of $D$ are orthogonal if every set (respec., subdigraph) $H \in \mathcal{H}$ contains exactly one vertex of $Q$. A well-known result of Gallai and Milgram shows that for every minimum path partition of a digraph there is a stable set orthogonal to it. Similarly, Gallai, Hasse, Roy and Vitaver independently proved that for every longest path of a digraph there is a vertex partition into stable sets (i.e, vertex-coloring) orthogonal to it. Berge showed that no analogous statements hold when optimality is required for the stable set or the vertex coloring. In this paper, we show that this holds if we replace stable sets by induced acyclic subdigraphs. In 1981, Linial proposed two generalizations of Gallai-Milgram and Gallai-Hasse-Roy-Vitaver results using a positive integer $k$ as a measure of optimality for the path partition and the coloring, respectively. These generalizations have led to two conjectures that remain open. Using the same strategy of replacing stable sets by induced acyclic subdigraphs, we prove relaxations of both conjectures.

翻译：设 $D$ 为有向图。$D$ 的不交顶点集族（或不交子图族）$\mathcal{H}$ 与 $D$ 的顶点子集（或子图）$Q$ 称为正交的，若每个集合（或子图）$H \in \mathcal{H}$ 恰好包含 $Q$ 中的一个顶点。Gallai 与 Milgram 的一个著名结果表明：对于有向图的每条最小路划分，存在一个稳定集与该划分正交。类似地，Gallai、Hasse、Roy 与 Vitaver 独立证明了：对于有向图的每条最长路，存在一个将顶点划分为稳定集的划分（即顶点染色）与该路正交。Berge 指出，当对稳定集或顶点染色要求最优性时，类似的结论不再成立。本文证明：若将稳定集替换为诱导无圈子图，则上述结论成立。1981 年，Linial 以正整数 $k$ 作为路划分与染色的最优性度量，分别提出了 Gallai-Milgram 结果与 Gallai-Hasse-Roy-Vitaver 结果的两个推广。这些推广引出了两个尚未解决的猜想。采用将稳定集替换为诱导无圈子图的相同策略，我们证明了这两个猜想的松弛形式。