k-Primitivity of digraphs

Let D be a directed graph (digraph) on n vertices. The digraph D is said to be primitive if for some m, between any ordered pair of vertices of D there is a directed walk of length m from the first vertex to the other. Here our focus is a generalization of primitivity, called k-primitivity, where k  -arc-colorings of digraphs are considered. Let kmax(n)kmax(n) be the maximum k for which there exists a k-coloring of some strong n-tournament that is k  -primitive. We show that (n−12)⩽kmax(n)<(n2)−⌈n4⌉.