{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:21:01Z","timestamp":1759335661995,"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>We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2\/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n\/d$ and $(2+o(1))\\sqrt{n\/\\log n}$ as $n\\to\\infty$. This is sharp up to constant factors.\u00a0Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\\min\\{\\sqrt{n},(n\\log n)\/d\\})$ as $n\\to\\infty$.\r\nRelatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\\sqrt{2}+o(1))\\sqrt{n\/\\log n}$ as $n\\to\\infty$. Second, any\u00a0 triangle-free graph on $n$ vertices has list chromatic number at most $O(\\sqrt{n\/\\log n})$ as $n\\to\\infty$.<\/jats:p>","DOI":"10.37236\/8650","type":"journal-article","created":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T02:20:07Z","timestamp":1590718807000},"source":"Crossref","is-referenced-by-count":4,"title":["Bipartite Induced Density in Triangle-Free Graphs"],"prefix":"10.37236","volume":"27","author":[{"given":"Wouter","family":"Cames van Batenburg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R\u00e9mi","family":"De Joannis de Verclos","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ross J.","family":"Kang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fran\u00e7ois","family":"Pirot","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,5,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p34\/8090","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p34\/8090","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T02:20:08Z","timestamp":1590718808000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,29]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/8650","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,5,29]]},"article-number":"P2.34"}}