Shen, Jian Directed triangles in digraphs. (English) Zbl 0904.05035 J. Comb. Theory, Ser. B 74, No. 2, 405-407 (1998). Let \(c\) be the smallest possible value such that every digraph on \(n\) vertices with minimum outdegree at least \(cn\) contains a directed triangle. It was conjectured by Caccetta and Häggkvist in 1978 that \(c=1/3\). Recently Bondy showed that \(c\leq (2\sqrt{6}- 3)/5= 0.3797\ldots\) by using some counting arguments. In this note, we prove that \(c\leq 3-\sqrt{7}= 0.3542\ldots\). Reviewer: Jian Shen (Kingston) Cited in 16 Documents MSC: 05C20 Directed graphs (digraphs), tournaments 05C38 Paths and cycles Keywords:digraph; directed triangle; minimum outdegree PDFBibTeX XMLCite \textit{J. Shen}, J. Comb. Theory, Ser. B 74, No. 2, 405--407 (1998; Zbl 0904.05035) Full Text: DOI Link References: [1] Behzad, M.; Chartrand, G.; Wall, C., On minimal regular digraphs with given girth, Fund. Math., 69, 227-231 (1970) · Zbl 0203.26502 [2] Bondy, J. A., Counting subgraphs: A new approach to the Caccetta-Häggkvist conjecture, Discrete Math., 165/166, 71-80 (1997) · Zbl 0872.05016 [3] L. Caccetta, R. Häggkvist, On minimal digraphs with given girth, Proceedings, Ninth S-E Conference on Combinatorics, Graph Theory and Computing, 1978, 181, 187; L. Caccetta, R. Häggkvist, On minimal digraphs with given girth, Proceedings, Ninth S-E Conference on Combinatorics, Graph Theory and Computing, 1978, 181, 187 · Zbl 0406.05033 [4] Li, Q.; Brualdi, R. A., On minimal regular digraphs with girth 4, Czechoslovak Math. J., 33, 439-447 (1983) · Zbl 0542.05035 [5] de Graaf, M.; Schrijver, A.; Seymour, P. D., Directed triangles in directed graphs, Discrete Math., 110, 279-282 (1992) · Zbl 0771.05045 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.