Open problems in gossip/broadcast schemes and the possible application of the method of local interchange

The gossip problem (telephone problem) is an information dissemination problem in which each of the n nodes of communication network has a unique piece of information that should be transmitted to all other nodes using two-way communications (telephone calls) between the pairs of nodes. During a call between the given two nodes, they exchange the whole information known to them at that moment. In this paper the method of local interchange is introduced to investigate gossip problems. This method is based on a repetitive use of permute higher and permute lower operations, which map one gossip graph with n vertices to another by moving only its edges without changing the labels of edges (the moments of corresponding calls). Using this operation we obtained results some of which are new and the others prove already the known ones in completely new way. However, the main topic of this paper is some open problems on the applications of this method.