Kernels of minimum size gossip schemes

The main part of gossip schemes are the kernels of their minimal orders. We give a complete characterization of all kernels that may appear in gossip schemes on simple graphs with a minimum number of calls. As consequences we prove several results on gossip schemes, e.g. the minimum number of rounds of a gossip scheme with a minimum number of calls is computed. Moreover, in the new context we give proofs of known results, e.g. the well-known Four-Cycle-Theorem. In the last part, we deal with order theoretic questions for such kernel posets. After describing all p-grid-kernels in terms of permutations and subsets, isomorphism is investigated and they are enumerated. Then we compute the order dimension and the jump number of all possible kernels , and nally, we show how to determine the numbers of their linear extensions.