Abstract
A 2-coloring ( G 1 , G 2 ) 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 ( G 1 , G 2 ) 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 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.
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