The structure of almost all graphs in a hereditary property

Abstract

A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n ↦ | P n | , where P n denotes the graphs of order n in P . It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then | P n | = 2 ( 1 − 1 / r + o ( 1 ) ) ( n 2 ) , where r = r ( P ) ∈ N is the so-called ‘colouring number’ of P . However, their results tell us very little about the structure of a typical graph G ∈ P . In this paper we describe the structure of almost every graph in a hereditary property of graphs, P . As a consequence, we derive essentially optimal bounds on the speed of P , improving the Alekseev–Bollobás–Thomason Theorem, and also generalising results of Balogh, Bollobás and Simonovits.