Exponents of 2-regular digraphs (Q1972142)

A digraph \(G\) is said to be primitive if for some positive integer \(k\), there is a walk of length exactly \(k\) from each vertex \(u\) to each vertex \(v\). The smallest such \(k\) in a primitive digraph \(G\) is called the exponent of \(G\). If each vertex of \(G\) has out-degree and in-degree exactly \(r\), then \(G\) is said to be \(r\)-regular. The paper shows that if \(G\) is a primitive 2-regular digraph with \(n\) vertices, then its exponent is not greater than \((n-1)^2/4+ 1\).