Assume a network (a complete undirected graph) comprised of $N$ participants (vertices). Each person holds a different piece of information, unknown to all other members. The participants can communicate in pairs, and with each communication, the two parties learn all the information that the other one knows. What is the least amount of such one-to-one exchanges, needed so that every participant knows everybody's information?
Some ideas that are not optimal:
The simplest solution is having everyone talk to everyone else (i.e. $N(N-1)$ communications).
A better idea is to have one participant collect everyone's information ($N - 1$ exchanges), and then redistribute it ($N - 2$ since the last one has already received everyone's information) for a total of $2N - 3$ exchanges.
Hint: Prove that the answer for $N = 5$ is $6$.
EDIT: The solution can be found in this MSE question: Minimal time gossip problem