Inducibility of Turán Graphs

• Inducibility of Turán graphs is defined as the limit of the maximum density of induced copies in large graphs, serving as a core measure in extremal graph theory.
• The analysis employs symmetrization techniques and continuous optimization to establish that balanced multipartite constructions uniquely maximize induced subgraph counts.
• Explicit formulas and stability results for almost balanced Turán graphs provide precise characterizations, even under K_(k+1)-free restrictions.

The inducibility of Turán graphs concerns the asymptotic maximum density of induced copies of a given Turán graph $F$ embedded within arbitrary large graphs, a principal theme in extremal graph theory. For a graph $F$, its inducibility is defined as

$$i(F)=\lim_{n\to\infty} \frac{I(F,n)}{\binom{n}{v(F)}}$$

where $I(F,n)$ denotes the maximum number of induced subgraphs isomorphic to $F$ across all $n$-vertex graphs. This problem admits a precise and complete solution for Turán graphs, especially in the regime where part sizes are almost balanced or differ by at most $O(\sqrt{a_r})$, with deep connections to symmetrization techniques, graphon optimization, and the theory of $K_{k+1}$-free graphs (Liu et al., 15 Jan 2026, Yuster, 18 Dec 2025).

1. Formal Definitions and Foundational Results

Let $F=T_r(a_1,\dots,a_r)$ denote a complete $r$-partite graph with part sizes $a_1 \geq \cdots \geq a_r \geq 1$, $v(F)=\ell=\sum_{i=1}^r a_i \geq r+1$. The function $I(F,G)$ counts the number of induced subgraphs of $G$ isomorphic to $F$; $I(F,n) = \max\{ I(F,G)\mid |V(G)|=n \}$.

The inducibility $i(F)$, introduced by Pippenger and Golumbic and further studied by Brown, Sidorenko, Bollobás, Egawa, Harris, Jin, Mubayi, Reiher, and Yuster, captures the sharp global upper bound for induced densities.

A Turán graph $T_m(n)$ is the complete $m$-partite graph on $n$ vertices whose part sizes differ by at most one. The central result for inducibility identifies $T_m(n)$ as extremal for $I(F,n)$ for all large enough $n$, with $m$ given explicitly by a discrete maximization problem: $$m = \arg\max_{k \geq r} f(k), \quad f(k)=\frac{(k-1)(k-2)\cdots (k-r+1)}{k^{\ell-1}}$$ where the unique maximizing $m$ is established by Lemma 2.1 (Liu et al., 15 Jan 2026).

2. Exact Inducibility Formulas for Turán Graphs

Theorem 1.2 yields the precise inducibility for almost balanced Turán graphs (i.e., $a_1<a_r+(1+\sqrt{8a_r+1})/2$ or $\binom{a_1-a_r}{2}<a_r$). Define the normalization constant

$$\kappa_F = \frac{\ell!}{\left[ \prod_{i=1}^r a_i! \right] \cdot \text{sym}(a_1,\dots,a_r)}$$

where $\text{sym}(a_1,\dots,a_r)$ is the number of automorphisms of the multiset of part-sizes.

For almost balanced $F$ with total size $\ell$ and $m$ as above,

$$i(F) = \kappa_F \cdot f(m) = \kappa_F \cdot \frac{(m-1)_{r-1}}{m^{\ell-1}}$$

This value is uniquely realized by $T_m(n)$ for $n \geq N_F$ for some $N_F$ dependent on $F$.

The explicit asymptotic count for large $n$ is: $$I(F,n) = I(F, T_m(n)) = \kappa_F \cdot (m-1)_{r-1}\frac{n^{\ell}}{m^{\ell-1}} + o(n^{\ell})$$

In the special case where $F=T(s,r)$ is the complete $r$-partite graph with $s$ vertices and part sizes as equal as possible (i.e., some parts of $p$, others $p+1$, $s=pr+q$), another equivalent formula emerges for $\ell = m$: $$i(F) = \frac{m! \, s!}{(m-r)! \, (r-q)! \, q! \, (p!)^{r} (p+1)^{q} m^{s}}$$ as systematically presented in (Yuster, 18 Dec 2025).

3. Optimization and Structure of Extremal Configurations

Reduction to complete multipartite graphs is achieved by Zykov symmetrization, as shown in Proposition 1 (Liu et al., 15 Jan 2026, Yuster, 18 Dec 2025), ensuring that for any extremal $G$, the maximum number of induced $F$-subgraphs can be realized by a complete multipartite structure.

The extremal problem transitions to a continuous optimization: for proportions $x \in \overline{P}$ (part-size weights), the inducibility can be recast as

$i(F) = \max_{x \in \overline{P}} p_F(x)$

where $p_F(x)$ is a symmetric polynomial encoding the induced density, and its maximizers (the "OPT sets") are shown to be balanced vectors: for almost balanced $F$, the unique maximizer is $(1/m, \dots, 1/m, 0, \dots, 0)$, confirmed via shifting inequalities (Proposition 3.3) and convexity analysis.

The resulting induced density $g(\ell)$ is unimodal in $\ell$ (Lemma 4.3 (Yuster, 18 Dec 2025)), increasing up to $m$ and then decreasing, thereby determining the unique maximizer.

4. $K_{k+1}$-Free Inducibility and Symmetrizable Families

A prominent extension addresses $I_{k+1}(F, n)$, the maximum number of induced copies of $F$ in a $K_{k+1}$-free $n$-vertex graph, with limiting inducibility

$$i_{k+1}(F)=\lim_{n\to\infty} \frac{I_{k+1}(F,n)}{\binom{n}{\ell}}$$

For almost balanced $F$, Theorem 1.4 (Liu et al., 15 Jan 2026) establishes: $$i_{k+1}(F)= \begin{cases} \kappa_F \cdot \frac{(k-1)_{r-1}}{k^{\ell-1}}, & \text{if } r \leq k < m\ \kappa_F \cdot \frac{(m-1)_{r-1}}{m^{\ell-1}}, & \text{if } k \geq m \end{cases}$$ The extremal construction is the $k$-partite Turán graph for $k < m$, and $m$-partite for $k \geq m$, responding to a conjecture of Bollobás–Egawa–Harris–Jin and a problem of Yuster (Yuster, 18 Dec 2025).

These methods apply more generally to symmetrizable families—the class of complete partite graphs for which induced densities in $K_k$-free graphs are maximized by multipartite configurations with number of parts $\ell < k$.

5. Extensions: Nearly Balanced Multipartite Graphs

Beyond strictly almost balanced graphs, the methodology extends to complete $r$-partite graphs $F=K_{a_1,\dots,a_r}$ with $a_1 - a_r = O(\sqrt{a_r})$. In this sub-$\sqrt{a_r}$ regime, optimality and stability results persist since key inequalities (notably the shifting inequality of Proposition 3.3, (Liu et al., 15 Jan 2026)) remain valid.

For these graphs, the unique maximizer of $p_F(x)$ remains the balanced weight vector. Perfect stability follows: any graph achieving inducibility close to the maximum must have edit distance $o(n^2)$ from the corresponding $m$-partite Turán graph (Theorem 1.5 (Liu et al., 15 Jan 2026)).

6. Explicit Calculations and Structural Implications

For $T(s,r)$ with $s \leq 3r+1$, explicit formulas for inducibility and the number of extremal parts are: $$t = \max\left\{ \ell \geq r \;\big|\; \frac{\ell}{\ell - r} \left( 1 - \frac{1}{\ell} \right)^{s} > 1 \right\}$$

$$i_k\bigl(T(s,r)\bigr) = \frac{\ell!\,s!}{(\ell-r)!\,(r-q)!\,q!\,(p!)^{r}\,(p+1)^{q}\,\ell^{s}}, \quad i\bigl(T(s,r)\bigr) = \frac{t!\,s!}{(t-r)!\,(r-q)!\,q!\,(p!)^{r}\,(p+1)^{q}\,t^{s}}$$

with $s=pr+q$, $0 \leq q < r$, $\ell = \min\{k-1, t\}$.

These results imply that for all Turán graphs with $s \leq 14$ (i.e., on up to 14 vertices), inducibility is completely determined; tables of values are compiled in (Yuster, 18 Dec 2025). The extremal configurations for the inducibility and $K_{k+1}$-free inducibility are always the balanced multipartite graphs with parts of (almost) equal size.

7. Proof Architectures and Stability

The proof strategy synthesizes classical symmetrization (Brown–Sidorenko), continuous optimization over the simplex of part proportions (Liu–Pikhurko), and stability results (Liu–Pikhurko–Sharifzadeh–Staden). Uniqueness and stability are established by analyzing local perturbations (removing/adding vertices/edges) and confirming that no alternative structure can improve, even asymptotically, upon $T_m(n)$ [(Liu et al., 15 Jan 2026), Theorem 1.5].

This line of analysis rigorously cements the $m$-partite Turán graph as uniquely extremal for inducibility across the natural families of complete multipartite (and in particular, Turán) graphs, for all sufficiently large $n$ and parameters within the quantified ranges.

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References

• (Liu et al., 15 Jan 2026) The inducibility of Turán graphs
• (Yuster, 18 Dec 2025) Inducibility in $H$-free graphs and inducibility of Turán graphs