{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T21:48:38Z","timestamp":1770328118841,"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>The 4 Color Theorem (4CT) implies that every\u00a0$n$-vertex planar graph has an independent set of size at least $\\frac{n}4$; this\u00a0is best possible, as shown by the disjoint union of many copies of $K_4$. \u00a0In 1968, Erd\u0151s\u00a0asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976\u00a0Albertson\u00a0showed (independently of the 4CT)\u00a0that every $n$-vertex planar graph has an independent set of size at least\u00a0$\\frac{2n}9$. Until now, this remained the best bound independent of the 4CT.\u00a0Our main result improves this bound to $\\frac{3n}{13}$.<\/jats:p>","DOI":"10.37236\/5309","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:25:09Z","timestamp":1578687909000},"source":"Crossref","is-referenced-by-count":5,"title":["Planar Graphs have Independence Ratio at least 3\/13"],"prefix":"10.37236","volume":"23","author":[{"given":"Daniel W.","family":"Cranston","sequence":"first","affiliation":[]},{"given":"Landon","family":"Rabern","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,9,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p45\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:16:53Z","timestamp":1579238213000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i3p45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,2]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,7,8]]}},"URL":"https:\/\/doi.org\/10.37236\/5309","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,9,2]]},"article-number":"P3.45"}}