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Some exact inducibility-type results for graphs via flag algebras

Levente Bodnár, Oleg Pikhurko

TL;DR

This work advances the graph edge-inducibility program by applying flag algebras to determine eleven new $\lambda_{\kappa,\ell}$ values for $\kappa \le 7$, together with perfect stability in all solved cases except $(4,3)$ where Erdős–Simonovits stability persists. The authors develop and deploy a comprehensive flag-algebra framework, including types, flags, and certificates, to bound induced subgraph densities and to characterise extremal blowup patterns. They also compute new $F$-inducibility constants for three 5-vertex graphs, revealing precise optimal constructions and stability properties, and provide detailed structural results for the $4$-cycle with a pendant edge. Notably, the $(4,3)$-problem exhibits a sharp ES-stable description but lacks perfect stability, leading to exact large-$n$ structural results distinguishing two extremal configurations; together with computational certificates, these findings illuminate the landscape of graph inducibility and its connections to size-forcible graphons and stability phenomena.

Abstract

The $(κ,\ell)$-edge-inducibility problem asks for the maximum number of $κ$-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 $κ\le 7$. 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.

Some exact inducibility-type results for graphs via flag algebras

TL;DR

This work advances the graph edge-inducibility program by applying flag algebras to determine eleven new values for , together with perfect stability in all solved cases except where Erdős–Simonovits stability persists. The authors develop and deploy a comprehensive flag-algebra framework, including types, flags, and certificates, to bound induced subgraph densities and to characterise extremal blowup patterns. They also compute new -inducibility constants for three 5-vertex graphs, revealing precise optimal constructions and stability properties, and provide detailed structural results for the -cycle with a pendant edge. Notably, the -problem exhibits a sharp ES-stable description but lacks perfect stability, leading to exact large- structural results distinguishing two extremal configurations; together with computational certificates, these findings illuminate the landscape of graph inducibility and its connections to size-forcible graphons and stability phenomena.

Abstract

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

Paper Structure

This paper contains 8 sections, 23 theorems, 72 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Flag algebras
  4. Perfect stability
  5. Computer-generated results
  6. Stability and exact result for the $(4,3)$-problem
  7. Structural results for the $4$-cycle plus a pendant edge
  8. Concluding remarks

Key Result

Lemma 2.1

Suppose that, for some $N$, we have a flag algebra certificate proving that $\lambda_{\gamma}\ifstrempty{}{}{()}\leqslant u$ as in eq:FAMain. Let $\tau$ be any type present in the certificate, say with $V(\tau)=[q]$, and let $s:=(N+q)/2$. (Thus $s$ is an integer and $s\geqslant q+1$). Then for every where the vector ${\bf v}_{(G,f)}^{\tau,s}$, as defined in eq:TauVector, lists the densities of $s$

Theorems & Definitions (27)

  • Lemma 2.1
  • Theorem 2.2: PikhurkoSliacanTyros19*Theorem 7.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Theorem 3.8
  • ...and 17 more