Berge's Conjecture and Aharoni-Hartman-Hoffman's Conjecture for locally in-semicomplete digraphs

TL;DR

This paper proves two longstanding conjectures related to path partitions and packings in a special class of directed graphs called locally in/out-semicomplete digraphs, advancing understanding in graph theory.

Contribution

It establishes the validity of Berge's and Aharoni-Hartman-Hoffman's conjectures specifically for locally in/out-semicomplete digraphs, a class of graphs with specific neighborhood properties.

Findings

Proves Berge's conjecture for locally in/out-semicomplete digraphs.

Proves Aharoni-Hartman-Hoffman's conjecture for the same class.

Extends the class of graphs for which these conjectures are known to hold.

Abstract

Let $k$ be a positive integer and let $D$ be a digraph. A path partition $\sP$ of $D$ is a set of vertex-disjoint paths which covers $V(D)$. Its $k$-norm is defined as $\sum_{P \in \sP} \Min{|V(P)|, k}$. A path partition is $k$-optimal if its $k$-norm is minimum among all path partitions of $D$. A partial $k$-coloring is a collection of $k$ disjoint stable sets. A partial $k$-coloring $\sC$ is orthogonal to a path partition $\sP$ if each path $P \in \sP$ meets $\min\{|P|,k\}$ distinct sets of $\sC$. Berge (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 $\sP$ is orthogonal to a coloring $\sC$ if each set $C \in \sC$ meets $\Min{|C|, k}$ paths of $\sP$. Aharoni, Hartman and Hoffman (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 $D$ is 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.