A variable version of the quasi-kernel conjecture

TL;DR

This paper explores variable versions of the quasi-kernel conjecture in digraphs, establishing implications between conjectures, proving new bounds for specific classes of digraphs, and answering open questions in the field.

Contribution

It shows the large QK conjecture implies the small QK conjecture with a weaker constant, proves equivalence of the large QK conjecture to a sharp version, and introduces variable conjectures with new bounds.

Findings

Large QK conjecture implies small QK conjecture with a weaker constant.

Large QK conjecture is equivalent to a sharp version.

Every sink-free digraph with bounded dichromatic number has a quasi-kernel of size at most (1-1/k)n.

Abstract

A quasi-kernel of a digraph $D$ is an independent set $Q$ such that every vertex can reach $Q$ in at most two steps. A 48-year conjecture made by P.L. Erd\H{o}s and Sz\'ekely, denoted the small QK conjecture, says that every sink-free digraph contains a quasi-kernel of size at most $n/2$. Recently, Spiro posed the large QK conjecture, that every sink-free digraph contains a quasi-kernel $Q$ such that $|N^-[Q]|\geq n/2$, and showed that it follows from the small QK conjecture. In this paper, we establish that the large QK conjecture implies the small QK conjecture with a weaker constant. We also show that the large QK conjecture is equivalent to a sharp version of it, answering affirmatively a question of Spiro. We formulate variable versions of these conjectures, which are still open in general. Not many digraphs are known to have quasi-kernels of size $(1-\alpha)n$ or less. We show this for digraphs with bounded dichromatic number, by proving the stronger statement that every sink-free digraph contains a quasi-kernel of size at most $(1-1/k)n$, where $k$ is the digraph's kernel-perfect number.