On the Girth of Digraphs with High Connectivity

In 1970, Behzad, Chartrand and Wall conjectured that the girth of every \(r\)-regular digraph \(G\) of order \(n\) is at most \(\left\lceil \frac{n}{r} \right\rceil\). The conjecture follows from a theorem of Menger and Dirac if \(G\) has strong connectivity \(x = r\). We show that any digraph with minimum in-degree and out-degree at least \(r\) has girth at most \(\left\lceil \frac{n}{r} \right\rceil\) if \(\kappa = r – 1\). We also find from the literature a family of counterexamples to a conjecture of Seymour.