Exponents of 2-regular digraphs

A digraph G is called primitive if for some positive integer k, there is a walk of length exactly k from each vertex u to each vertex v (possibly u again). If G is primitive, the smallest such k is called the exponent of G, denoted by exp(G). A digraph G is said to be r-regular if each vertex in G has outdegree and indegree exactly r. It is proved that if G is a primitive 2-regular digraph with n vertices, then exp(G) (n?1) 2 4 + 1. Also all 2-regular digraphs with exponents attaining the bound are characterized. This supports a conjecture made by Shen and Gregory.