Say that a tournament T has 2-property, if for every distinct vertices $u,w \in V(T)$, T has a (directed) u,w-path of length exactly 2. (In particular, if T has 2-property, then every vertex of T is a king. But not every tournament in which each vertex is a king has 2-property.) Determine the least positive integer n $\geq$ 2 such that some n-vertex tournament T has 2-property. (Hint: show that each in-degree and out-degree in such tournament must be at least 3, and then present a construction.)
I'm confused on the idea that 2-property would mean every vertex is a king but not that a tournament with each vertex is a king has 2-property. Does this mean we can't have u,w path of length of 1 at all but every path should be exactly two? How do we tackle this? Do we just add loops to each vertex? Please help! Thanks!