Abstract: A three-colored digraph D is primitive if and only if there exists nonnegative integers h, k and v with h+k+v>0 such that for each pair (i,j) of vertices there exists a (h,k,v)-walk in D from i to j. The exponent of the primitive three-colored digraph D is the minimum value of　h+k+v taken over all such h, k and v. Special threecolored digraphs were studied, whose uncolored digraph consists of one n-cycle, one (n-1)-cycle and one 2-cycle. Some primitive conditions and an upper bound on the exponent were given. Further, the characterizations of extremal three-colored digraphs were put forth.

Key words: extremal digraph , primitive exponent, three-colored digraph