On the exact maximum induced density of almost all graphs and their inducibility

TL;DR

This paper determines the maximum number of induced copies of almost all graphs in large graphs, characterizes extremal structures, and refines bounds on the inducibility of typical graphs, revealing that most graphs have inducibility close to a known lower bound.

Contribution

It provides an explicit characterization of graphs with maximum induced copies for almost all graphs and establishes that random graphs satisfy this property with high probability.

Findings

For almost all graphs H, the maximum induced copies in graphs of size n are exactly given by a specific formula for n up to 2^{√h}.

Extremal graphs achieving this maximum are characterized as balanced blowups of H with some internal edges.

The inducibility of typical graphs H is shown to be close to the lower bound, with an asymptotic formula.

Abstract

Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices. Let $f(n,h) = \Pi_{i}^h a_i$ where $\sum_{i=1}^h a_i = n$ and the $a_i$ are as equal as possible. Let $g(n,h) = f(n,h) + \sum_{i=1}^h g(a_i,h)$. It is proved that for almost all graphs $H$ on $h$ vertices it holds that $i_H(n)=g(n,h)$ for all $n \le 2^{\sqrt{h}}$. More precisely, we define an explicit graph property ${\cal P}_h$ which, when satisfied by $H$, guarantees that $i_H(n)=g(n,h)$ for all $n \le 2^{\sqrt{h}}$. It is proved, in particular, that a random graph on $h$ vertices satisfies ${\cal P}_h$ with probability $1-o_h(1)$. Furthermore, all extremal $n$-vertex graphs yielding $i_H(n)$ in the aforementioned range are determined. We also prove a stability result. For $H \in {\cal P}_h$ and a graph $G$ with $n \le 2^{\sqrt{h}}$ vertices satisfying $i_H(G) \ge f(n,h)$, it must be that $G$ is obtained from a balanced blowup of $H$ by adding some edges inside the blowup parts. The {\em inducibility} of $H$ is $i_H = \lim_{n \rightarrow \infty} i_H(n)/\binom{n}{h}$. It is known that $i_H \ge h!/(h^h-h)$ for all graphs $H$ and that a random graph $H$ satisfies almost surely that $i_H \le h^{3\log h}h!/(h^h-h)$. We improve upon this upper bound almost matching the lower bound. It is shown that a graph $H$ which satisfies ${\cal P}_h$ has $i_H =(1+O(h^{-h^{1/3}}))h!/(h^h-h)$.