The inducibility of blow-up graphs

TL;DR

This paper proves that graphs with the maximum number of induced copies of large balanced blow-ups of a fixed graph are structurally close to being blow-ups of that graph, providing an asymptotic characterization.

Contribution

It establishes that extremal graphs maximizing induced copies of large balanced blow-ups are essentially blow-ups of the original graph, answering a longstanding open question.

Findings

Graphs with maximum induced copies resemble blow-ups of the original graph

Provides an asymptotic characterization of extremal structures

Answers a question posed in prior research (BEHJ95)

Abstract

The blow-up of a graph is obtained by replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. If every vertex is replaced with the same number of copies, then the resulting graph is called a balanced blow-up. We show that any graph which contains the maximum number of induced copies of a sufficiently large balanced blow-up of H is itself essentially a blow-up of H. This gives an asymptotic answer to a question in [BEHJ95].