The Scrambling Index of Two-colored Wielandt

Abstract A digraph is primitive provided there is a positive integer k such that for each pair of vertices u and v there exist walks of length k from u to v and from v to u. The scrambling index of a primitive digraph D is the smallest positive integer k such that for each pair of vertices u and v in D there is a vertex w such that there exist walks of length k from u to w and from v to w. A two-colored digraph is a digraph each of whose arc is colored by red or blue. In this paper we generalize the notion of scrambling index of a primitive digraph to that of two-colored digraph. We define the scrambling index of a two-colored digraph D(2) to be the smallest positive integer h+ ℓ over all pairs of nonnegative integers (h, ℓ) such that for each pair of distinct vertices u and v there is a vertex w with the property that there are walks form u to w and from v to w consisting of h red arcs and ℓ blue arcs. For two-colored Wielandt digraph on n 4 vertices we show the scrambling index lies on the interval [n2 3n+ 3, n2 2n+ 2]