A Class of Primitive Two-Colored Digraph with Large Competition Index

Abstract

The competition index of a primitive two-colored digraph D^2, denoted k(D^((2))), is the smallest positive integer h+l such that for each pair of vertices u and v there is vertex w with the property that there is a (h,l)-walk from v to w. For two-colored digraph on n vertices it is known that k(D^((2) ))≤(3n^3+2n^2-2n)/2. In this work, we discuss a class of primitive two-colored digraph consisting of two cycles whose scrambling index closes to (3n^3+2n^2-2n)/2