A class of solutions to the gossip problem, part I

We further develop the corresp0ndenc.e established in Part I of this paper to characterize optimal soluticns to the gossip problem in which no one hears his own information. To reiterate, these are graphs on n vertices with 2n -4. edges (n even), for which the edges c:ln bc linearly ordered to produce an increasing path from each vertex to every other, without having an imzzeasing path from any vertex ted itself. With two exceptions, we call these NOHO-graphs. In PaTt I, we associated each NOHO-graph with a quadruple consisting of two permutations and two binary sequences. In Part IP, we study the properties OF the quadruples associated with NOIHO-graphs. We characterize the quadruples that can .arist:. The main result is that any pair of Fequences in such a quadruple uniquely determines the other pair. Using this, we count the realizable quadruples and several subclasses of them. There are 3(n-6)‘2 realizable quadruples associated with NOHO-graphs on II vertices. The subclasses enumerated have various symmetry properties; their sizes are powers of 2 and 3. In IPart II we also study the properties of a combining operator that we define for NOHO-graphs and their associated quadruples. In Part III, we Gil use these properties to enumerate the non-isomorphic NOHO-graphs.