{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:52Z","timestamp":1753893832440,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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\u0151s and Sz\u00e9kely, known as the small QK conjecture, says that every sink-free digraph contains a quasi-kernel of size at most $n\/2$.\r\nRecently, Spiro posed the large QK conjecture, that every digraph contains a quasi-kernel $Q$ such that $|N^-[Q]|\\geq n\/2$, and showed that it follows from the small QK conjecture.\r\nIn 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.\r\nNot many digraphs are known to have quasi-kernels of size $(1-\\alpha)n$ or less. We show that digraphs with bounded dichromatic number have quasi-kernels of size at most $(1-\\alpha)n$, by proving a stronger statement.<\/jats:p>","DOI":"10.37236\/13653","type":"journal-article","created":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T15:18:58Z","timestamp":1750346338000},"source":"Crossref","is-referenced-by-count":0,"title":["A Variable Version of the Quasi-Kernel Conjecture"],"prefix":"10.37236","volume":"32","author":[{"given":"Jiangdong","family":"Ai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiangzhou","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fei","family":"Peng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,6,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p53\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T15:18:58Z","timestamp":1750346338000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i2p53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,20]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,4,11]]}},"URL":"https:\/\/doi.org\/10.37236\/13653","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,6,20]]},"article-number":"P2.53"}}