On directed triangles in digraphs

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 α ≥ 0.35312, then each n-vertex digraph D with minimum outdegree at least an has a directed β-cycle. If β ≥ 0.34564, then every n-vertex digraph D in which the outdegree and the indegree of each vertex is at least βn has a directed 3-cycle.