Exponents of a class of special three-colored primitive digraphs with $n$ vertices in graph theory

The exponent problems of the three-colored digraphs containing $n$ vertices, one $n$-cycle, one $(n-3)$-cycle, and one $3$-cycle are considered. According to the coloring of the $3$-cycle, we only consider the case where there are zero red arcs, one yellow arc, and two blue arcs in the 3-cycle. The primitive conditions of the cycle matrix are given, based on the discussion of different cases, the upper bound of the primitive exponents is found, and the extreme digraphs are characterized. The results are useful for the study of the primitive exponents of three-colored digraphs in general cases and the application of graph coloring problems.