Given a directed graph with V vertices, what is the minimum number of (directed) edges (no self-loops) for a cycle of length L to necessarily exist.
I can show that we can add 1 directed edge for every pair of vertex without creating a cycle (it becomes a directed acyclic graph). It is easy to argue that, after this exercise, the next directed edge will create a loop. Therefore :
$\frac{V(V-1)}{2}+1$ directed edges in a graph imply existence of some cycle.
How do I move ahead from here?
Edit: I am assuming that at most 1 directed edge can exist between any two vertices.