The structure of almost all graphs in a hereditary property

TL;DR

This paper characterizes the typical structure of graphs within hereditary properties, leading to improved bounds on their growth rates and generalizing previous structural and enumerative results.

Contribution

It provides a detailed description of the structure of almost all graphs in hereditary properties, refining existing bounds and extending prior structural theorems.

Findings

Almost all graphs in a hereditary property have a specific structural form.

The bounds on the speed of hereditary properties are essentially optimal.

The results generalize and improve previous theorems by Balogh, Bollobás, and Simonovits.

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 \mapsto |\P_n|, where \P_n denotes the graphs of order n in \P. It was shown by Alekseev, and by Bollobas and Thomason, that if \P is a hereditary property of graphs then |\P_n| = 2^{(1 - 1/r + o(1))n^2/2}, where r = r(\P) \in \N is the so-called `colouring number' of \P. However, their results tell us very little about the structure of a typical graph G \in \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-Bollobas-Thomason Theorem, and also generalizing results of Balogh, Bollobas and Simonovits.