On directed triangles in digraphs
Summary: Using a recent result of \textit{M. Chudnovsky}, \textit{P. Seymour}, and \textit{B. Sullivan} [''Cycles in dense digraphs'', Combinatorica 28, No.\,1, 1--18 (2008; Zbl 1174.05049)], we slightly improve two bounds related to the Caccetta-Häggkvist Conjecture [\textit{L. Caccetta} and \textit{R. Häggkvist}, ''On minimal digraphs with given girth'', Congressus Numerantium XXI, 181--187 (1978; Zbl 0406.05033) ]. Namely, we show that if \(\alpha\geq 0.35312\), then each \(n\)-vertex digraph \(D\) with minimum outdegree at least \(\alpha n\) has a directed 3-cycle. If \(\beta\geq 0.34564\), then every \(n\)-vertex digraph \(D\) in which the outdegree and the indegree of each vertex is at least \(\beta n\) has a directed 3-cycle.
scientific article; zbMATH DE number 5540819