A two-colored digraph D is primitive if there exist nonnegative integers hand k with h + k > 0 such that for each pair (i, j) of vertices there exists an (h, k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is the minimum value of h + k taken over all such hand k. In this paper, we give the tight upper bound on the exponents of a class of primitive two-colored digraphs with (s + 1) n-cycles and one (n - 1)-cycle, and the characterizations of the extremal two-colored digraphs.