Some exact inducibility-type results for graphs via flag algebras

TL;DR

This paper applies flag algebra methods to solve inducibility problems for graphs, determining maximum induced subgraph counts and structures for large graphs in several new cases involving small subsets.

Contribution

It introduces new exact results for inducibility problems in graphs using flag algebras, including solutions for all large graphs in eleven cases with small subsets and structure results for three specific 5-vertex graphs.

Findings

Resolved inducibility for all large graphs in eleven new cases with small subsets.

Computed asymptotic maximum densities for induced copies of three specific 5-vertex graphs.

Described extremal and near-extremal graph structures for these inducibility problems.

Abstract

The $(\kappa,\ell)$-edge-inducibility problem asks for the maximum number of $\kappa$-subsets inducing exactly $\ell$ edges that a graph of given order $n$ can have. Using flag algebras and stability approach, we resolve this problem for all sufficiently large $n$ (including a description of all extremal and almost extremal graphs) in eleven new non-trivial cases when $\kappa\le 7$. We also compute the $F$-inducibility constant (the asymptotically maximum density of induced copies of $F$ in a graph of given order $n$) and obtain some corresponding structure results for three new graphs $F$ with $5$ vertices: the 3-edge star plus an isolated vertex, the 4-cycle plus an isolated vertex, and the 4-cycle with a pendant edge.