A two-colored digraph D is primitive if there exist nonnegative integers h and k with h+k0 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 defined as the minimum value of h+k taken over all such h and k.The tight upper bounds and lower bounds of exponents of a class of special and primitive two-colored digraphs with two cycles was given

and give the characterizations of one type extremal two-colored digraphs was given.