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Directed triangles in digraphs. (English) Zbl 0904.05035

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 \(c\leq 3-\sqrt{7}= 0.3542\ldots\).

MSC:

05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
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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
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[5] de Graaf, M.; Schrijver, A.; Seymour, P. D., Directed triangles in directed graphs, Discrete Math., 110, 279-282 (1992) · Zbl 0771.05045
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