{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:48Z","timestamp":1753893828223,"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>A colouring of a digraph as defined by Neumann-Lara in 1982 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be its dichromatic number. This quantity has been widely studied in the last decades and can be considered as a natural directed analogue of the chromatic number of a graph. A digraph $D$ is called even if for every $0$-$1$-weighting of the edges it contains a directed cycle of even total weight. We show that every non-even digraph has dichromatic number at most $2$ and an optimal colouring can be found in polynomial time. We strengthen a previously known NP-hardness result by showing that deciding whether a directed graph is $2$-colourable remains NP-hard even if it contains a feedback vertex set of bounded size.<\/jats:p>","DOI":"10.37236\/8800","type":"journal-article","created":{"date-parts":[[2022,10,18]],"date-time":"2022-10-18T01:22:13Z","timestamp":1666056133000},"source":"Crossref","is-referenced-by-count":0,"title":["Colouring Non-Even Digraphs"],"prefix":"10.37236","volume":"29","author":[{"given":"Marcelo Garlet","family":"Millani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Raphael","family":"Steiner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sebastian","family":"Wiederrecht","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2022,10,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i4p9\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i4p9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,18]],"date-time":"2022-10-18T01:22:13Z","timestamp":1666056133000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i4p9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,7]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,10,6]]}},"URL":"https:\/\/doi.org\/10.37236\/8800","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,10,7]]},"article-number":"P4.9"}}