Typical $T$-free graphs

TL;DR

This paper characterizes the structure of almost all graphs that do not contain a given tree T as an induced subgraph, revealing a partition into parts with specific properties and deriving bounds on T-free graphs.

Contribution

It proves that for each non-edge tree T, almost every T-free graph can be partitioned into parts with $P_4$-free induced subgraphs, providing bounds on the number of T-free graphs.

Findings

Partition of T-free graphs into $ ext{α}(T)-1$ parts with specific properties

Bounds on the number of T-free graphs, often tight up to a constant

Almost every T-free graph has chromatic number equal to its largest clique size

Abstract

We prove that for every tree $T$ which is not an edge, for almost every graph $G$ which does not contain $T$ as an induced subgraph, $V(G)$ has a partition into $\alpha(T)-1$ parts certifying this fact. Each part induces a graph which is $P_4$-free and has further properties which depend on $T$. As a consequence we obtain good bounds (often tight up to a constant factor) on the number of $T$-free graphs and show in a follow-up paper~\cite{RY} that almost every $T$-free graph $G$ has chromatic number equal to the size of its largest clique.