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

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## 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)=Π i h a i where ∑ i=1 h a i =n and the a i are as equal as possible. Let g(n,h)=f(n,h)+∑ 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≤2 h . More precisely, we define an explicit graph property P h which, when satisfied by H, guarantees that i H (n)=g(n,h) for all n≤2 h . It is proved, in particular, that a random graph on h vertices satisfies 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∈P h and a graph G with n≤2 h vertices satisfying i H (G)≥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 inducibility of H is i H =lim n→∞ ⁡i H (n)/(nh). It is known that i H ≥h!/(h h −h) for all graphs H and that a random graph H satisfies almost surely that i H ≤h 3log⁡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 P h has i H =(1+O(h −h 1/3 ))h!/(h h −h).

Original language English 81-109 29 Journal of Combinatorial Theory. Series B 136 https://doi.org/10.1016/j.jctb.2018.09.005 Published - May 2019

## Keywords

• Induced density
• Inducibility
• Packing

## ASJC Scopus subject areas

• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
• Computational Theory and Mathematics

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