Paths and stability number in digraphs

TL;DR

This paper proves a conjecture that for any stability number k, there exists a digraph where removing any k-1 directed paths still leaves a digraph with the same stability number, demonstrating the theorem's optimality.

Contribution

It confirms Hahn and Jackson's 1990 conjecture, establishing the sharpness of the Gallai-Milgram theorem for all positive integers k.

Findings

Confirmed the conjecture for all k

Constructed specific digraphs with the property

Showed the theorem's bounds are tight

Abstract

The Gallai-Milgram theorem says that the vertex set of any digraph with stability number k can be partitioned into k directed paths. In 1990, Hahn and Jackson conjectured that this theorem is best possible in the following strong sense. For each positive integer k, there is a digraph D with stability number k such that deleting the vertices of any k-1 directed paths in D leaves a digraph with stability number k. In this note, we prove this conjecture.