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 3logh 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 |
---|---|
Pages (from-to) | 81-109 |
Number of pages | 29 |
Journal | Journal of Combinatorial Theory. Series B |
Volume | 136 |
DOIs | |
State | Published - May 2019 |
Bibliographical note
Publisher Copyright:© 2018 Elsevier Inc.
Keywords
- Induced density
- Inducibility
- Packing
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics