Primitive exponent of a class of two-colored digraphs with two cycles

A two-colored digraph D is primitive if there exist nonnegative integers h and 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 special two-colored digraphs whose uncolored digraph has m + n vertices and consists of one m-cycle and one n-cycle are considered. Some primitive conditions and the upper bound of the exponents are given. The characterizations of extremal two-colored digraphs are also given.