{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:45:37Z","timestamp":1759063537154,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gy\u00e1rf\u00e1s and proved that if a graph $G$ has no odd holes then $\\chi(G)\\le 2^{2^{\\omega(G)+2}}$. Chudnovsky, Robertson, Seymour and Thomas showed that if $G$ has neither $K_4$ nor odd holes then $\\chi(G)\\le 4$. In this note, we show that if a graph $G$ has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then $\\chi(G)\\le 4$ and $\\chi(G)\\le 3$ if $G$ has radius at most $3$, and for each vertex $u$ of $G$, the set of vertices of the same distance to $u$ induces a bipartite subgraph. This answers some questions in Plummer and Zha (2014).<\/jats:p>","DOI":"10.37236\/5555","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:43:18Z","timestamp":1578670998000},"source":"Crossref","is-referenced-by-count":9,"title":["A Note on Chromatic Number and Induced Odd Cycles"],"prefix":"10.37236","volume":"24","author":[{"given":"Baogang","family":"Xu","sequence":"first","affiliation":[]},{"given":"Gexin","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Xiaoya","family":"Zha","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,11,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p32\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:42:29Z","timestamp":1579236149000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i4p32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,3]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/5555","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,11,3]]},"article-number":"P4.32"}}