Exponents of vertex-transitive digraphs

A digraph D is primitive if, for some positive integer r, there is a u ! v walk of length r for every pair u; v of vertices of D. The minimum such r is called the exponent of D, denoted exp(D). We prove that if a primitive digraph D of order n is an Abelian Cayley digraph with degree k 4, then exp(D) max n 5; n dk=2e 1 o : If we assume only that the primitive digraph D is vertex-transitive, then we are able to prove that exp(D) n k n k + 1 : Finally, we make a conjecture on the exponents of all regular primitive digraphs. Supported by an NSERC Postdoctoral Fellowship. Supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP0005134.