Abstract: In order to study the primitive exponent problem of nonnegative matrix pairs, the two-colored digraphs are extended to three-colored digraphs. Using the corresponding relation between a digraph and a matrix, a class of primitive three-colored digraphs whose uncolored digraphs has n vertices, consists of one n-cycle and two (n－1)-cycles is studied. Some primitive conditions, a tight upper bound of primitive exponents and the characterization of extremal three-colored digraphs are given.

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