Exponents of 2-colorings of loopless, symmetric digraphs

Abstract

A 2-coloring (G 1, G 2) is primitive if there exist non-negative integers h and k with h + k > 0 such that for each pair (u, v) of vertices there exists an (h, k)-walk in G from u to v. The exponent of (G 1, G 2) is the minimum value of h + k taken over all such h and k. In this article, we consider 2-colorings of loopless, symmetric digraphs give the conditions for a 2-coloring of loopless, symmetric digraph to be primitive and establish an upper bound on the exponents.