Orthogonality between acyclic subdigraphs and paths in digraphs

TL;DR

This paper explores the orthogonality properties between acyclic subdigraphs and paths in digraphs, extending classical results and proposing relaxations of open conjectures using acyclic structures.

Contribution

It demonstrates that replacing stable sets with induced acyclic subdigraphs preserves certain orthogonality properties and provides relaxations for two open conjectures.

Findings

Gallai-Milgram and Gallai-Hasse-Roy-Vitaver results extend to acyclic subdigraphs

Proves relaxations of two conjectures involving path partitions and colorings

Shows orthogonality properties hold with acyclic subdigraphs, not just stable sets

Abstract

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.