{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:00Z","timestamp":1753893840200,"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>Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\\chi$ contains a rainbow independent set of size $\\lceil\\frac12\\chi\\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy.\r\nDrawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $\\chi$ contains an induced cycle of length $\\Omega(\\chi\\log\\chi)$ as $\\chi\\to\\infty$. Even if one only demands an induced path of length $\\Omega(\\chi\\log\\chi)$, the conclusion would be sharp up to a constant multiple. We prove it for regular girth $5$ graphs and for girth $21$ graphs.\r\nAs a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some $c &gt;0$ such that for every forest $H$ on $D$ vertices, every triangle-free and induced $H$-free graph has chromatic number at most $c D\/\\log D$. We prove this assertion with 'triangle-free' replaced by 'regular girth 5'.<\/jats:p>","DOI":"10.37236\/9267","type":"journal-article","created":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T01:56:49Z","timestamp":1625018209000},"source":"Crossref","is-referenced-by-count":0,"title":["Structure and Colour in Triangle-Free Graphs"],"prefix":"10.37236","volume":"28","author":[{"given":"N. R.","family":"Aravind","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stijn","family":"Cambie","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wouter","family":"Cames van Batenburg","sequence":"additional","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":"Viresh","family":"Patel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,6,18]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i2p47\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i2p47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T01:56:49Z","timestamp":1625018209000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i2p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,18]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/9267","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,6,18]]},"article-number":"P2.47"}}