{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:46:07Z","timestamp":1759063567731,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Brooks' Theorem states that a connected graph $G$ of maximum degree $\\Delta$ has chromatic number at most $\\Delta$, unless $G$ is an odd cycle or a complete graph. A result of Johansson shows that if $G$ is triangle-free, then the chromatic number drops to $O(\\Delta \/ \\log \\Delta)$. In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph $D$ without directed cycles of length two has chromatic number $\\chi(D) \\leq (1-e^{-13}) \\tilde{\\Delta}$, where $\\tilde{\\Delta}$ is the maximum geometric mean of the out-degree and in-degree of a vertex in $D$, when $\\tilde{\\Delta}$ is sufficiently large. As a corollary it is proved that there exists an absolute constant $\\alpha  &lt;  1$ such that $\\chi(D) \\leq \\alpha (\\tilde{\\Delta} + 1)$ for every $\\tilde{\\Delta} &gt; 2$.<\/jats:p>","DOI":"10.37236\/682","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:41:22Z","timestamp":1578714082000},"source":"Crossref","is-referenced-by-count":10,"title":["Strengthened Brooks' Theorem for Digraphs of Girth at least Three"],"prefix":"10.37236","volume":"18","author":[{"given":"Ararat","family":"Harutyunyan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bojan","family":"Mohar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,10,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p195\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p195\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:01:18Z","timestamp":1579302078000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p195"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,3]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/682","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,10,3]]},"article-number":"P195"}}