Implicit representations and factorial properties of graphs

The idea of implicit representation of graphs was introduced in Kannan et al. (1992) and can be defined as follows. A representation of an nn-vertex graph GG is said to be implicit if it assigns to each vertex of GG a binary code of length O(logn)O(logn) so that the adjacency of two vertices is a function of their codes. Since an implicit representation of an nn-vertex graph uses O(nlogn)O(nlogn) bits, any class of graphs admitting such a representation contains 2O(nlogn)2O(nlogn) labelled graphs with nn vertices. In the terminology of Balogh et al. (2000) such classes have at most factorial speed of growth. In this terminology, the implicit graph conjecture can be stated as follows: every class with at most factorial speed of growth which is hereditary admits an implicit representation. The question of deciding whether a given hereditary class has at most factorial speed of growth is far from being trivial. In the present paper, we introduce a number of tools simplifying this question. Some of them can be used to obtain a stronger conclusion on the existence of an implicit representation. We apply our tools to reveal new hereditary classes with the factorial speed of growth. For many of them we show the existence of an implicit representation.