3055 2020 eng 1 27 article The Electronic Journal of Combinatorics 0 -- 2025-07-17 -- The Electronic Journal of Combinatorics 1077-8926 10.37236/8942 true {"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,12]],"date-time":"2025-07-12T01:23:52Z","timestamp":1752283432940},"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>The dichromatic number $\\overrightarrow{\\chi}(D)$\u00a0of a digraph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class induces an acyclic subdigraph of $D$. A digraph $D$ is $k$-critical if $\\overrightarrow{\\chi}(D) = k$ but $\\overrightarrow{\\chi}(D') &lt; k$ for all proper subdigraphs $D'$ of $D$. We examine methods for creating infinite families of critical digraphs, the Dirac join and the directed and bidirected Haj\u00f3s join. We prove that a digraph $D$ has dichromatic number at least $k$ if and only if it contains a subdigraph that can be obtained from bidirected complete graphs on $k$ vertices by directed Haj\u00f3s joins and identifying non-adjacent vertices. Building upon that, we show that a digraph $D$ has dichromatic number at least $k$ if and only if it can be constructed from bidirected $K_k$'s by using directed and bidirected Haj\u00f3s joins and identifying non-adjacent vertices (so called Ore joins), thereby transferring a well-known result of Urquhart to digraphs. Finally, we prove a Gallai-type theorem that characterizes the structure of the low vertex subdigraph of a critical digraph, that is, the subdigraph, which is induced by the vertices that have in-degree $k-1$ and out-degree $k-1$ in $D$.<\/jats:p>","DOI":"10.37236\/8942","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:44:15Z","timestamp":1584578655000},"source":"Crossref","is-referenced-by-count":7,"title":["Haj\u00f3s and Ore Constructions for Digraphs"],"prefix":"10.37236","volume":"27","author":[{"given":"J\u00f8rgen","family":"Bang-Jensen","sequence":"first","affiliation":[]},{"given":"Thomas","family":"Bellitto","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Schweser","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Stiebitz","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,3,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p63\/8050","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p63\/8050","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:44:15Z","timestamp":1584578655000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p63"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8942","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,20]]}}} journal-article crossref PersonAuthorFirstName_1,PersonAuthorLastName_1,PersonAuthorFirstName_2,PersonAuthorLastName_2,PersonAuthorFirstName_3,PersonAuthorLastName_3,PersonAuthorFirstName_4,PersonAuthorLastName_4,PublisherName,TitleMain_1,TitleAbstract_1,TitleParent_1,Issue,Volume,CompletedYear,IdentifierIssn doi-import false false 2 Jørgen Bang-Jensen Thomas Bellitto Thomas Schweser Michael Stiebitz