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 δ and girth g contains a directed path of length δ(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 δ and girth g contains a directed path of 2δ(1-2/g). For small g we obtain better bounds, e.g. for $g=3$ we show that oriented graph with minimum out-degree δ contains a directed path of length 1.5δ. Furthermore, we show that each d-regular digraph with girth g contains a directed path of length Ω(dg/log d). Our results give the first non-trivial bounds for these problems.