Berge’s Conjecture and Aharoni–Hartman–Hoffman’s Conjecture for Locally In-Semicomplete Digraphs

Abstract

Let k be a positive integer and let D be a digraph. A path partition $$\mathcal {P}$$ of D is a set of vertex-disjoint paths which covers V(D). Its k-norm is defined as $$\sum _{P \in \mathcal {P}} \min \{|V(P)|, k\}$$ . A path partition is k-optimal if its k-norm is minimum among all path partitions of D. A partialk-coloring is a collection of k disjoint stable sets. A partial k-coloring $$\mathcal {C}$$ is orthogonal to a path partition $$\mathcal {P}$$ if each path $$P \in \mathcal {P}$$ meets $$\min \{|V(P)|,k\}$$ distinct sets of $$\mathcal {C}$$ . Berge (Eur J Comb 3(2):97–101, 1982) conjectured that every k-optimal path partition of D has a partial k-coloring orthogonal to it. A (path) k-pack of D is a collection of at most k vertex-disjoint paths in D. Its weight is the number of vertices it covers. A k-pack is optimal if its weight is maximum among all k-packs of D. A coloring of D is a partition of V(D) into stable sets. A k-pack $$\mathcal {P}$$ is orthogonal to a coloring $$\mathcal {C}$$ if each set $$C \in \mathcal {C}$$ meets $$\min \{|C|, k\}$$ paths of $$\mathcal {P}$$ . Aharoni et al. (Pac J Math 2(118):249–259, 1985) conjectured that every optimal k-pack of D has a coloring orthogonal to it. A digraph D is semicomplete if every pair of distinct vertices of D are adjacent. A digraph D is locally in-semicomplete if, for every vertex $$v \in V(D)$$ , the in-neighborhood of v induces a semicomplete digraph. Locally out-semicomplete digraphs are defined similarly. In this paper, we prove Berge’s and Aharoni–Hartman–Hoffman’s Conjectures for locally in/out-semicomplete digraphs.