On the length of directed paths in digraphs

Thomass\'e conjectured the following strengthening of the well-known Caccetta-Haggkvist Conjecture: any digraph with minimum out-degree and girth g contains a directed path of length (g-1). Bai and Manoussakis gave counterexamples to Thomass\'e's conjecture for every even g 4. In this note, we first generalize their counterexamples to show that Thomass\'e's conjecture is false for every g 4. We also obtain the positive result that any digraph with minimum out-degree and girth g contains a directed path of 2 (1-2g). For small g we obtain better bounds, e. g. for g=3 we show that oriented graph with minimum out-degree contains a directed path of length 1. 5. Furthermore, we show that each d-regular digraph with girth g contains a directed path of length (dg/ d). Our results give the first non-trivial bounds for these problems.