Local exponents of a class of two-colored digraphs consisting of two cycles with lengths 2s+1 and s

A two-colored digraph D⁽²⁾ is a digraph each of whose arcs is colored by red or blue. An (h,k)-walk is a walk of length h + k consisting of h red arcs and k blue arcs. A two-colored digraph is primitive provided there exist nonnegative integers h and k such that for each pair of vertices u and v there is an (h,k)-walk from u to v. The local exponent of a vertex v in D⁽²⁾, denoted by exp(D⁽²⁾,v), is the smallest positive integer h + k over all pairs of nonnegative integers h and k such that for every vertex u in D⁽²⁾ there is an (h,k)-walk from u to v. For integer s ≥ 5, we present local exponents of two-colored digraphs D⁽²⁾ consisting of two cycles whose lengths 2s + 1 and s, respectively, and has one vertex in common. For any vertex v in the two-colored digraph D⁽²⁾ we present the exp(D⁽²⁾,v) in terms of s, the distances from the terminal vertices of the blue arcs to the vertex of indegree 2, and the distance from the vertex of indegree 2 to the vertex v.