Enumeration of Full Graphs: Onset of the Asymptotic Region

A full graph on n vertices, as deened by Fulkerson, is a representation of the intersection and containment relations among a system of n sets. It has an undirected edge between vertices representing intersecting sets, and a directed edge from a to b if the corresponding set A contains B. Kleitman, Lasaga and Cowen gave a uniied argument to show that asymptotically, almost all full graphs can be obtained by taking an arbitrary undirected graph in the n vertices, distinguishing a clique in this graph which need not be maximal, and then adding directed edges going out from each vertex in the clique to all vertices to which there is not already an existing undirected edge. Call graphs of this type members of the dominant class. This paper obtains the rst upper and lower bounds on how large n has to be, so that the asymptotic behavior is indeed observed. It is shown that when n > 170, the dominant class predominates, while when n < 17, the full graphs in the dominant class comprise less than half of the total number of realizable full graphs on n vertices.