Local Exponent of Asymmetric Two-colored Lollipops

A two-colored digraph is a digraph each of whose arcs is colored by red or blue. An (h,k)-walk in a two-colored digraph is a walk consisting of h red arcs and k blue arcs. A two-colored digraph is primitive provided that for each pair of vertices a and nu there exists an (h,k)-walk from a to nu and from nu to a. The smallest of such positive integer h+k is called the exponent of D. Let D be a primitive two colored digraph and let nu be a vertex in D. The local exponent off) at the vertex nu, denoted exp(nu,D), is defined to be the smallest positive integer h+k over all nonnegative integers h and k such that for each vertex u in D there exists an (h,k)-walk from nu to a. An (s,n s)-lollipop is a connected symmetric digraphs on n vertices consisting of an s-cycle and an (n-s)-path with one vertex in common. A two-colored lollipop is called asymmetric provided that the arc (u,nu) is colored red whenever (nu,u) is blue and vice versa. We discuss local exponent of asymmetric two-colored lollipop especially the class (n,0)-lollipop where n is odd. We present formula for local exponent of a vertex nu that depend on n and the distance of nu to a special vertex that lies on the n-cycle.