Upper Bound on the Primitive Exponents of a Class of Three-Colored Digraphs

Abstract

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 this paper, a class of special 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.