Upper Bound of Primitive Exponents of a Class of Two-colored Digraph with n Vertices

Abstract

A two-colored directed digraph D is primitive if and only if there exist nonnegative integers h and k with h+k>0 such that for each pair (i, j) of vertices there is a (h, k)-walk in D from i to j. A (h, k)-walk from i to j consisting of h red arcs and k blue arcs. The exponent of the primitive two-colored digraph D, denoted exp(D), is defined to be the smallest value of h+k over all such h and k. A class of two-colored digraphs with two cycles whose uncolored digraph has n vertices and consists of one n-cycle and one 3-cycle is considered. The upper bound of primitive exponent and characteristic of extremal two-colored digraphs are given.