Upper bound on the primitive exponents of a class of three-colored digraphs

摘要

A three-colored digraph D is primitive if and only if there exist nonnegative integers h, k and v with h + k + v > 0 such that for each pair (i, j) of vertices there is an (h, k, v)-walk in D from i to j. The exponent of the primitive three-colored digraph D is defined to be the smallest value of h + k + v over all such h,k and v. In the paper, a class of especial primitive three-colored digraphs with n vertices, consisting of one n-cycle and two (n - 1)-cycles, are considered. For the case a = c - 1, some primitive conditions, the tight upper bound on the exponents and the characterization of extremal three-colored digraphs are given.