{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T21:12:26Z","timestamp":1774645946712,"version":"3.50.1"},"reference-count":13,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2025,2,4]],"date-time":"2025-02-04T00:00:00Z","timestamp":1738627200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[2025,7]]},"abstract":"<jats:title>ABSTRACT<\/jats:title><jats:p>One of the most fundamental results in graph theory is Mantel's theorem which determines the maximum number of edges in a triangle\u2010free graph of order . Recently, a colorful variant of this problem has been solved. In this variant we consider  graphs on a common vertex set, think of each graph as edges in a distinct color, and want to determine the smallest number of edges in each color which guarantees the existence of a rainbow triangle. Here, we solve the analogous problem for directed graphs without rainbow triangles, either directed or transitive, for any number of colors. The constructions and proofs essentially differ for  and  and the type of the forbidden triangle. Additionally, we also solve the analogous problem in the setting of oriented graphs.<\/jats:p>","DOI":"10.1002\/jgt.23224","type":"journal-article","created":{"date-parts":[[2025,2,5]],"date-time":"2025-02-05T05:17:54Z","timestamp":1738732674000},"page":"269-281","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Directed Graphs Without Rainbow Triangles"],"prefix":"10.1002","volume":"109","author":[{"given":"Sebastian","family":"Babi\u0144ski","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science Jagiellonian University Krak\u00f3w Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2770-7180","authenticated-orcid":false,"given":"Andrzej","family":"Grzesik","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science Jagiellonian University Krak\u00f3w Poland"}]},{"given":"Magdalena","family":"Prorok","sequence":"additional","affiliation":[{"name":"AGH University Krakow Poland"}]}],"member":"311","published-online":{"date-parts":[[2025,2,4]]},"reference":[{"key":"e_1_2_11_2_1","article-title":"A Rainbow Version of Mantel's Theorem","author":"Aharoni R.","year":"2020","journal-title":"Advances in Combinatorics"},{"key":"e_1_2_11_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/22M1535048"},{"key":"e_1_2_11_4_1","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.21189"},{"key":"e_1_2_11_5_1","unstructured":"A.DiwanandD.Mubayi Tur\u00e1n's Theorem With Colors (2007). preprint http:\/\/www.math.cmu.edu\/~mubayi\/papers\/webturan.pdf."},{"key":"e_1_2_11_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.23109"},{"key":"e_1_2_11_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-024-00102-6"},{"key":"e_1_2_11_8_1","unstructured":"P.Frankl \u201cGraphs Without Rainbow Triangles \u201darXiv:2203.07768(2022)."},{"key":"e_1_2_11_9_1","doi-asserted-by":"publisher","DOI":"10.37236\/12784"},{"key":"e_1_2_11_10_1","doi-asserted-by":"publisher","DOI":"10.1112\/blms.12896"},{"key":"e_1_2_11_11_1","doi-asserted-by":"publisher","DOI":"10.37236\/11676"},{"key":"e_1_2_11_12_1","doi-asserted-by":"publisher","DOI":"10.1112\/blms.12343"},{"key":"e_1_2_11_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2003.08.005"},{"key":"e_1_2_11_14_1","doi-asserted-by":"publisher","DOI":"10.37236\/4615"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.23224","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T05:16:05Z","timestamp":1747199765000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.23224"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,4]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["10.1002\/jgt.23224"],"URL":"https:\/\/doi.org\/10.1002\/jgt.23224","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,4]]},"assertion":[{"value":"2023-08-09","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-01-10","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-02-04","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}