Exponents of two-colored digraphs consisting of two cycles

A two-colored digraph D⁽²⁾ is a digraph D whose each of its arc is colored by either red or blue. For nonnegative integers s and t with s+t > 0, an (s, t)-walk in a two-colored digraph is a walk of length s+t consisting of s red arcs and t blue arcs. The vector (s, t)^T is the composition of the walk. A two-colored digraph D⁽²⁾ is primitive provided that there exist nonnegative integers h and k such that for each pair of vertices u and v in D⁽²⁾ there is an (h, k)-walk in D⁽²⁾ from u to v. For a two-colored digraph consisting of two cycles D⁽²⁾, we established a formula for the exponent of D⁽²⁾ in terms of the compositions of its cycles and the compositions of two specified paths in D⁽²⁾. We then apply the formula to the class of two-colored digraphs consisting of two cycles whose lengths differ by 1 and to the class of two-colored digraph consisting a cycle and a loop.