{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:25:33Z","timestamp":1768292733977,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $D=(V,A)$ be a digraph. A vertex set $K\\subseteq V$ is a quasi-kernel 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, Chv\u00e1tal and Lov\u00e1sz proved that every digraph has a quasi-kernel. P. L. Erd\u0151s and L. A. Sz\u00e9kely 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).<\/jats:p>","DOI":"10.37236\/11043","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T07:46:15Z","timestamp":1662623175000},"source":"Crossref","is-referenced-by-count":3,"title":["Towards the Small Quasi-Kernel Conjecture"],"prefix":"10.37236","volume":"29","author":[{"given":"Alexandr V.","family":"Kostochka","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruth","family":"Luo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Songling","family":"Shan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2022,9,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i3p49\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i3p49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T07:46:15Z","timestamp":1662623175000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i3p49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,9]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,7,1]]}},"URL":"https:\/\/doi.org\/10.37236\/11043","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,9]]},"article-number":"P3.49"}}