Exponents of 2-coloring of symmetric digraphs

A 2-coloring (G₁,G₂) of a digraph is 2-primitive if there exist nonnegative integers h and k with h+k>0 such that for each ordered pair (u,v) of vertices there exists an (h,k)-walk in (G₁,G₂) from u to v. The exponent of (G₁,G₂) is the minimum value of h+k taken over all such h and k. In this paper, we consider 2-colorings of strongly connected symmetric digraphs with loops, establish necessary and sufficient conditions for these to be 2-primitive and determine an upper bound on their exponents. We also characterize the 2-colored digraphs that attain the upper bound and the exponent set for this family of digraphs on n vertices.