Typical structure of hereditary graph families. I. Apex-free families

TL;DR

This paper refines the understanding of the typical structure of hereditary graph families, especially apex-free families, and characterizes those with speeds just above a known threshold, advancing graph theory classification.

Contribution

It provides a more precise structural description for a class of hereditary families and generalizes previous results on hereditary graph family speeds.

Findings

Characterization of hereditary families with speed just above the threshold

Refined description of typical graphs in apex-free hereditary families

Extension of previous classifications of hereditary graph families

Abstract

A family of graphs $\mathcal{F}$ is hereditary if $\mathcal{F}$ is closed under isomorphism and taking induced subgraphs. The speed of $\mathcal{F}$ is the sequence $\{|\mathcal{F}^n|\}_{n \in \mathbb{N}}$, where $\mathcal{F}^n$ denotes the set of graphs in $\mathcal{F}$ with the vertex set $[n]$. Alon, Balogh, Bollob\'{a}s and Morris [The structure of almost all graphs in a hereditary property, JCTB 2011] gave a rough description of typical graphs in a hereditary family and used it to show for every proper hereditary family $\mathcal{F}$ there exist $\varepsilon>0$ and an integer $l \geq 1$ such that $$|\mathcal{F}^n| = 2^{(1-1/l)n^2/2+o(n^{2-\varepsilon})}.$$ The main result of this paper gives a more precise description of typical structure for a restricted class of hereditary families. As a consequence we characterize hereditary families with the speed just above the threshold $2^{(1-1/l)n^2/2}$, generalizing a result of Balogh and Butterfield [Excluding induced subgraphs: Critical graphs, RSA 2011].