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 $\delta$ and girth $g$ contains a directed path of length $\delta(g-1)$. Bai and Manoussakis \cite{Bai} gave counterexamples to Thomass\'{e}'s conjecture for every even $g\geq 4$. In this note, we first generalize their counterexamples to show that Thomass\'{e}'s conjecture is false for every $g\geq 4$. We also obtain the positive result that any digraph with minimum out-degree $\delta$ and girth $g$ contains a directed path of $2(1-\frac{2}{g})$. For small $g$ we obtain better bounds, e.g.~for $g=3$ we show that oriented graph with minimum out-degree $\delta$ contains a directed path of length $1.5\delta$. Furthermore, we show that each $d$-regular digraph with girth $g$ contains a directed path of length $\Omega(dg/\log d)$. Our results give the first non-trivial bounds for these problems.