A variable version of the quasi-kernel conjecture

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 Erd{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]|≥ 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-α)n or less. We show this for digraphs with bounded dichromatic number, by proving that every sink-free digraph contains a quasi-kernel of size at most kn/k+1, where k is the dichromatic number of D.