On coloring the arcs of a tournament, covering shortest paths, and reducing the diameter of a graph

We define closed edge colorings of directed graphs, and state a conjecture about the maximum size of a tournament graph that can be arc-colored with mm colors and contain no closed subgraphs. We prove special cases of this conjecture. We show that if this conjecture is correct then for any (undirected) graph with positive edge lengths and a given subset V′V′ of nodes, covering all the shortest paths between pairs of nodes of V′V′ requires at least |V′|−1|V′|−1 edges. We use the latter property to produce an approximation algorithm with improved bound for minimizing the diameter or the radius of an unweighted graph by adding to it a given number of new edges.