Towards the Small QuasiKernel Conjecture
Abstract
Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasikernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chvátal and Lovász proved that every digraph has a quasikernel. P. L. Erdős and L. A. Székely in 1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$ has a quasikernel of size at most $V/2$. This conjecture is only confirmed for narrow classes of digraphs, such as semicomplete multipartite, quasitransitive, or locally demicomplete 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 4colorable graphs (in particular, of all planar graphs).
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.04003
 Bibcode:
 2020arXiv200104003K
 Keywords:

 Mathematics  Combinatorics