Toward optimal gossiping schemes with conference calls

n people have distinct bits of information which they can communicate by k-person conference calls. The object of gossip is to inform everyone of everyone else's information. Which networks admit a minimum call gossip scheme? For k = 2 a necessary and sufficient condition for such a network is that it is connected and contains a cycle of length 4. We address the same question for k>2. The values of n are partitioned into 2 classes: trivial and nontrivial. A necessary and sufficient condition is given for networks of size n for trivial n. For nontrivial n a sufficient condition is given and its necessity is conjectured.