A note on the girth of digraphs

Behzad, Chartrand and Wall conjectured that the girth of a diregular graph of ordern and outdegreer is not greater than n /r]. This conjecture has been proved forr=2 by Behzad and forr=3 by Bermond. We prove that a digraph of ordern and halfdegree ≧4 has girth not exceeding n / 4]. We also obtain short proofs of the above results. Our method is an application of the theory of connectivity of digraphs.