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 $D$ 为一个有向图。$D$ 的一组不相交顶点集（或不相交子图）$\mathcal{H}$ 与 $D$ 的一个顶点子集（或子图）$Q$ 称为正交的，若每个集合（或子图）$H \in \mathcal{H}$ 恰包含 $Q$ 的一个顶点。Gallai 与 Milgram 的一个著名结果表明，对于有向图的每个最小路径划分，存在一个与之正交的稳定集。类似地，Gallai、Hasse、Roy 与 Vitaver 独立证明，对于有向图的每条最长路径，存在一个将顶点划分为稳定集（即顶点着色）的划分与之正交。Berge 指出，当要求稳定集或顶点着色最优时，类似结论不成立。本文表明，若将稳定集替换为诱导无圈子图，则上述结论成立。1981年，Linial 基于正整数 $k$（作为路径划分与着色最优性的度量）提出了 Gallai-Milgram 定理与 Gallai-Hasse-Roy-Vitaver 定理的两个推广，这些推广引出了两个仍待解决的猜想。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。