A jump to the bell number for hereditary graph properties

Abstract

A hereditary graph property is a collection of labeled graphs, closed under isomorphism and also under the taking of induced subgraphs. Its speed is the number of graphs in the property as a function of the number of vertices in the graph. Earlier research has characterized the speeds for hereditary graph properties up to n(1+o(1))n, and described the properties that have those smaller speeds. The present work provides the minimal speed possible above that range, and gives a structural characterization for properties which exhibit such speeds.More precisely, this paper sheds light on the jump from below n(1+o(1))n to the range that includes n(1+o(1))n. A measure jumps when there are two functions with positive distance such that the measure can take no values between those functions. A clean jump occurs when the bounding functions are well-defined and occur as possible values of the measure. It has been known for some time that the density of a graph jumps; recent work on hereditary graph properties has shown that speeds jump for properties with “large” or “small” speeds.The current work shows that there is a clean jump for properties with speed in a middle range. In particular, we show that when the speed of a hereditary graph property has speed greater than ncn for all c<1, the speed is at least Bn, the nth Bell number. Equality occurs only for the property containing all disjoint unions of cliques or its complement.