  In the theory of digraphs, the study of cycles is a subject of great
importance and has given birth to a number of deep questions such as the
Behzad-Chartrand-Wall conjecture (1970) and its generalization, the
Caccetta-H\"{a}ggkvist conjecture (1978). Despite a lot of interest and
efforts, the progress on these remains slow and mostly restricted to the
solution of some special cases. In this note, we prove these conjectures for
digraphs with girth is at least as large as their minimum out-degree and
without short even cycles. More generally, we prove that if a digraph has
sufficiently large girth and does not contain closed walks of certain lengths,
then the conjectures hold. The proof makes use of some of the known results on
the Caccetta-H\"{a}ggkvist conjecture, properties of direct products of
digraphs and a construction that multiplies the girth of a digraph.
