Reordered gossip schemes

This paper presents a new theorem called the reordering theorem for the problem of gossiping by telephone. The theorem says the calls of any complete gossiping can be rearranged into a sequence such that at most two calls cause each participant to become fully informed, and such that the reverse of this sequence has the same property. This theorem has broad consequences. Several results from the literature are easy corollaries, including the four-cycle theorem and Burosch, Leont'ev, and Markosyan's structure theorem for label-connected graphs. The reordering theorem also provides a solution to the problem of gossiping with minimum cost, which has been unsolved in the literature for more than fteen years.