Exponents of a class of two-colored digraphs

Abstract

A two-colored digraph D is primitive if there exist nonnegative integers h and k with h+k>0 such that for each pair (i, j) of vertices there exists an (h, k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is the minimum value of h+k taken over all such h and k. In this article, we consider special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consist of one n-cycle and one (n − 2)-cycle. We give the bounds on the exponents, and the characterizations of the extremal two-colored digraphs.