Primitive digraphs with smallest large exponent

Abstract

A primitive digraph D on n vertices has large exponent if its exponent, γ ( D ) , satisfies α n ⩽ γ ( D ) ⩽ w n , where α n = ⌊ w n / 2 ⌋ + 2 and w n = ( n - 1 ) 2 + 1 . It is shown that the minimum number of arcs in a primitive digraph D on n ⩾ 5 vertices with exponent equal to α n is either n + 1 or n + 2 . Explicit constructions are given for fixed n even and odd, for a primitive digraph on n vertices with exponent α n and n + 2 arcs. These constructions extend to digraphs with some exponents between α n and w n . A necessary and sufficient condition is presented for the existence of a primitive digraph on n vertices with exponent α n and n + 1 arcs. Together with some number theoretic results, this gives an algorithm that determines for fixed n whether the minimum number of arcs is n + 1 or n + 2 .