Scrambling index of two-colored two-cycles whose lengths differ by four

Abstract

A two-cycles is a digraph consists of two cycles. A two-colored digraph is a digraph whose arcs are colored by red or blue. Scrambling index of a primitive two-colored digraph D(2), denoted by k(D(2)), is the smallest positive integer h + ℓ over all pairs of nonnegative integers h and ℓ such that for every pair of vertices u and v in D(2) there is a vertex w in D(2) with the property that there exist a walk from u to w and a walk from v to w consisting of h red arcs and ℓ blue arcs. We discuss the scrambling index of a certain class of primitive two-colored two-cycles with cycles of length c and c + 4, respectively, where c ≡ 1(mod 4). We present a formula for k(D(2)) that depends on c and the distance of two certain vertices to the vertex of indegree 2.