  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
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 $2\delta(1-\frac{1}{g})$. 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.
