Inducibility in $H$-free graphs and inducibility of Turán graphs
Raphael Yuster
TL;DR
The paper advances a systematic theory of inducibility in H-free graphs, showing that for symmetrizable families the inducibility over H-free graphs reduces to a simpler χ(H)-bounded problem and is realized by complete multipartite graphons. It provides a detailed analysis for Turán graphs, proving that many cases are governed by a single equi-partite graphon W with a computable explicit formula, and it extends the framework to broader complete multipartite graphs, including conditions under which the extremal graphon has distinct part sizes. A striking takeaway is that almost all F are inducibility diverse, and that for Turán-type graphs the inducibility can be determined up to small vertex-bound cases, yielding complete determinations for all Turán graphs on at most 14 vertices. The work yields new insights into when inducibility is attained by simple structured graphons and highlights interesting phenomena where i(F) is rational but i_k(F) can be irrational, underscoring rich arithmetic behavior in extremal densities.
Abstract
For graphs $F$ and $H$, let $i(F)$ denote the inducibility of $F$ and let $i_H(F)$ denote the inducibility of $F$ over $H$-free graphs. We prove that for almost all graphs $F$ on a given number of vertices, $i_{K_k}(F)$ attains infinitely many values as $k$ varies. For complete partite graphs $F$ (and, more generally, for symmetrizable families of graphs $F$), we prove that $i_H(F)=i_{K_k}(F)$ where $k=χ(H)$, and is attained by a complete $\ell$-partite graphon $W_{F,k}$, where $\ell < k$. We determine the part sizes of $W_{F,k}$ for all $k$, whence determine $i(F)$, whenever $F$ is the Turán graph on $s$ vertices and $r$ parts, for all $s \le 3r+1$, which was recently proved by Liu, Mubayi, and Reiher for $s=r+1$. As a corollary, this determines the inducibility of all Turán graphs on at most $14$ vertices. Furthermore, since inducibility is invariant under complement, this determines the inducibility of all matchings and, more generally, all graphs with maximum degree $1$, of any size. Similarly, this determines the inducibility of all triangle factors, of any size. For complete partite graphs $F$ with at most one singleton part, we prove that $i_{K_k}(F)$ only attains finitely many values as $k$ varies; in particular, there exists $t=t(F)$ such that $i(F)$ is attained by some complete $t$-partite graphon. This is best possible as it was shown by Liu, Pikhurko, Sharifzadeh, and Staden that this is not necessarily true if there are two singleton parts. Finally, for every $r$, we give a nontrivial sufficient condition for a complete $r$-partite graph $F$ to have the property that $i(F)$ is attained by a complete partite graphon all whose part sizes are distinct.