Kernels of minimum size gossip schemes Original Research Article

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 finally, we show how to determine the numbers of their linear extensions.