Upper generalized exponents of two-colored primitive extremal ministrong digraphs

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 u and ν there exists an (h,k)-walk from u to ν and from ν to u. The smallest of such positive integer h+k is called the exponent of D⁽²⁾. Let D⁽²⁾ be a primitive two colored digraph and let X be a nonempty subset of V(D⁽²⁾). The set exponent of X, denoted exp(X,D), is defined to be the smallest positive integer h+k over all nonnegative integers h and k such that for each vertex ν in D⁽²⁾ there exists an (h,k)-walk from at least one vertex in X to ν. The kth upper generalized exponent of D, F(k,D⁽²⁾) is defined to be $F(k,D(2)) = max{exp(X,D(2)):|X| = k}$⁠. We discuss upper generalized exponents of two-colored primitive extremal ministrong digraphs on n vertices, that is a two-colored digraph consisting of an (n-1)-cycle and an (n-2)-cycle. We present a lower bound for the kth upper generalized exponent that depends on n and k.