Towards the Small Quasi-Kernel Conjecture - Kostochka, Alexandr V.; Luo, Ruth; Shan, Songling (The Electronic Journal of Combinatorics, 2022)
Let D=(V,A) be a digraph. A vertex set KV is a quasi-kernel of D if K is an independent set in D and for every vertex vVK, v is at most distance 2 from K. In 1974, Chvátal and Lovász proved that every digraph has a quasi-kernel. P. L. Erdős and L. A. Székely in 1976 conjectured that if every vertex of D has a positive indegree, then D has a quasi-kernel of size at most |V|/2. This conjecture is only confirmed for narrow classes of digraphs, such as semicomplete multipartite, quasi-transitive, or locally semicomplete digraphs. In this note, we state a similar conjecture for all digraphs, show that the two conjectures are equivalent, and prove that both conjectures hold for a class of digraphs containing all orientations of 4-colorable graphs (in particular, of all planar graphs).