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.