Exponents of a class of two-colored digraphs with two cycles

A two-colored digraph is a digraph whose arcs are colored red or blue. A two-colored digraph is primitive provided that there exist nonnegative integers h and k with h+k>0 such that for each pair (i,j) of vertices there is an (h,k)-walk from i to j in D. The exponent of D is the minimum value of h+k taken over all such h and k. In this paper, we consider a class of special primitive two-colored digraphs whose uncolored digraphs have n+s vertices and consist of one n-cycle and one (n-t)-cycle for t⩾1. We give bounds on the exponents and characterize the extreme two-colored graphs, which generalizes the results in [Y. Gao, Y. Shao, Exponents of two-colored digraphs with two cycles, Linear Algebra Appl. 407 (2005) 263–276; Y. Gao, Y. Shao, Exponents of a class two-colored digraphs, Linear and Multilinear Algebra 53(3) (2005) 175–188].