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{\bf Peter Hamburger, Penny Haxell and Alexandr Kostochka}
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{\bf On Directed Triangles in Digraphs}
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Using a recent result of Chudnovsky, Seymour, and Sullivan, we
slightly improve two bounds related to the Caccetta-Haggkvist
Conjecture. 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.
\bye