Sidorenko’s Conjecture in Extremal Graph Theory

Updated 7 July 2026

Sidorenko's Conjecture is a statement in extremal graph theory asserting that every bipartite graph H attains its minimum homomorphism density in a quasirandom host graph or graphon.
It connects analytic graphon methods with combinatorial proofs, being verified for classes like trees, even cycles, complete bipartite graphs, and hypercubes.
Advanced techniques such as tree decompositions, entropy arguments, and group-theoretic reductions are employed to deepen our understanding of quasirandomness and graph limits.

Searching arXiv for recent and foundational papers on Sidorenko's conjecture. Sidorenko’s conjecture is a central conjecture in extremal graph theory and graph limits asserting that for every bipartite graph $H$ , the homomorphism density of $H$ into a host graph or graphon is minimized, at fixed edge density, by the quasirandom host. In graphon form, for a symmetric measurable $W:[0,1]^2\to[0,1]$ , it states

$t_H(W)\ge t_{K_2}(W)^{e(H)},$

and in finite-graph form, for a graph $G$ with edge density $p$ ,

$t_H(G)\ge p^{e(H)}.$

Equivalently, the Erdős–Rényi random graph asymptotically minimizes the number of $H$ -homomorphisms among graphs of the same edge density (Kim et al., 2013). The conjecture is only formulated for bipartite $H$ ; the analogous universal inequality fails outside the bipartite setting (Zhao, 13 Jun 2026).

1. Formulations and basic meaning

Sidorenko’s conjecture has two standard equivalent formulations. For a bipartite graph $H=(A\cup B,E(H))$ and a bounded, non-negative, symmetric, measurable function $H$ 0, the analytic form is

$H$ 1

In graphon language, if

$H$ 2

then the conjecture is precisely $H$ 3 (Kim et al., 2013).

For finite graphs, with

$H$ 4

Sidorenko’s conjecture is

$H$ 5

Since $H$ 6, the conjecture says that among graphs with fixed edge density, the Erdős–Rényi model asymptotically minimizes the $H$ 7-homomorphism count (Kim et al., 2013).

This perspective places the conjecture at the interface of extremal combinatorics, quasirandomness, and graph limits. In the graphon setting, the constant graphon is the conjectured minimizer. In the finite setting, the conjecture predicts that edge correlations cannot reduce the density of a fixed bipartite pattern below the independent-edge benchmark (Conlon et al., 2010).

2. Historical position and verified classes

The conjecture is attributed to Sidorenko and, in a closely related form, to Erdős and Simonovits (Li et al., 2011). The paper literature summarized here records several classical verified families: paths, trees, even cycles, complete bipartite graphs, bipartite graphs with at most four vertices on one side, and hypercubes (Kim et al., 2013). The hypercube case was established via norming-graph methods, while trees and cycles admit more classical Hölder- and Cauchy–Schwarz-type proofs (Conlon et al., 2010).

A major exact result concerns bipartite graphs with a universal vertex on one side. If $H$ 8 is bipartite and some vertex of $H$ 9 is adjacent to all vertices of $W:[0,1]^2\to[0,1]$ 0, then $W:[0,1]^2\to[0,1]$ 1 satisfies Sidorenko’s conjecture (Conlon et al., 2010). This class was later rederived by a short logarithmic-calculus argument, and the same framework proved the forcing conjecture for such graphs when they are not trees (Li et al., 2011).

Approximate forms are also known. If $W:[0,1]^2\to[0,1]$ 2 is bipartite with width $W:[0,1]^2\to[0,1]$ 3, then

$W:[0,1]^2\to[0,1]$ 4

with equality of exponents when $W:[0,1]^2\to[0,1]$ 5, i.e. precisely in the universal-vertex case (Conlon et al., 2010). This gives a uniform lower bound for all bipartite graphs, but it is weaker than the full conjecture whenever $W:[0,1]^2\to[0,1]$ 6.

The landscape of verified classes has continued to expand through structural decomposition results, product theorems, subdivision theorems, local graphon inequalities, and reductions to highly symmetric hosts (Conlon et al., 2015). At the same time, the paper literature repeatedly emphasizes that the conjecture remains very open, and even small regular bipartite graphs such as $W:[0,1]^2\to[0,1]$ 7 historically served as benchmark unknown cases (Kim et al., 2013).

3. Structural methods: tree-arrangeability, tree decompositions, and biregularity

One influential structural approach is tree-arrangeability. For a bipartite graph $W:[0,1]^2\to[0,1]$ 8 with bipartition $W:[0,1]^2\to[0,1]$ 9, write $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 0 for the neighborhood of $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 1. An independent set $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 2 is $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 3-arrangeable for a tree $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 4 on $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 5 if for every pair $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 6 the path $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 7 from $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 8 to $t_H(W)\ge t_{K_2}(W)^{e(H)},$ 9 satisfies

$G$ 0

If a bipartition exists with $G$ 1 tree-arrangeable, then $G$ 2 has Sidorenko’s property (Kim et al., 2013). This covers, among other examples, complete bipartite graphs, the universal-vertex class, and graphs with two dominating neighborhood-maximal vertices on one side (Kim et al., 2013).

The same paper identifies tree-arrangeability with a specific tree-decomposition condition: $G$ 3 is tree-arrangeable if and only if there is a tree decomposition whose bags are $G$ 4 for $G$ 5 (Kim et al., 2013). This links Sidorenko’s conjecture to tree decompositions and conditional independence structures resembling Markov random fields.

That perspective was generalized by strong tree decompositions and then by higher tree decompositions. Strongly tree-decomposable graphs were shown to satisfy Sidorenko’s conjecture via a branching-random-walk embedding and entropy argument (Conlon et al., 2015). Higher tree decompositions iterate this construction: a graph is $G$ 6-strongly tree-decomposable if it admits a recursive tree decomposition with induced-forest overlaps and compatible lower-level decompositions, and every such graph satisfies Sidorenko’s conjecture (Conlon et al., 2018).

A different reduction concerns the host rather than the pattern. It is enough to verify Sidorenko’s inequality on biregular bigraphons, meaning bigraphons with almost-everywhere constant left and right degree functions equal to the edge density (Coregliano et al., 2021). That reduction was originally due to Szegedy; a later paper gave an elementary proof and used the same regularization ideas to derive further results, including reflective tree decompositions. A bigraph admitting a reflective tree decomposition with a core that weakly dominates each intersection graph is Sidorenko; this framework unifies strong tree decompositions and $G$ 7-decompositions (Coregliano et al., 2021).

These approaches share a common pattern: they replace an arbitrary graph by a recursive assembly of pieces for which the entropy, conditional-expectation, or degree-profile calculations can be controlled. This suggests that much of the known positive theory is organized around decomposability of either the pattern or the host.

4. Analytic and probabilistic frameworks

A second major line of attack is analytic. The logarithmic calculus of Li and Szegedy uses Jensen’s inequality for $G$ 8 and for $G$ 9 to derive subgraph-density inequalities directly from graphon integrals (Li et al., 2011). In its simplest form it recovers the path inequalities behind the Blakley–Roy theorem, and it yields a concise proof for the universal-vertex class (Li et al., 2011). The same paper introduces a strengthened notion of smoothness for labeled subgraphs, shows that smoothness is stable under gluing, and derives new Sidorenko families called reflection trees (Li et al., 2011).

Szegedy later developed an information-theoretic formulation. For finite graphs, if $p$ 0 denotes the uniform distribution on $p$ 1 and $p$ 2 the uniform measure on all maps $p$ 3, then

$p$ 4

so Sidorenko’s conjecture becomes an entropy inequality. This framework builds witness measures by conditionally independent couplings and leads to the class of thick graphs, every one of which is Sidorenko (Szegedy, 2014). The same paper extends the method to large classes of $p$ 5-uniform hypergraphs, even though the naive hypergraph analogue of Sidorenko’s conjecture is false in general (Szegedy, 2014).

There is also a local analytic theory. Lovász proved a local form of Sidorenko’s conjecture: if a graphon $p$ 6 is sufficiently close to the constant graphon, then every bipartite graph $p$ 7 satisfies $p$ 8 (Lovász, 2010). One of the paper’s central inequalities states that if $p$ 9 has girth $t_H(G)\ge p^{e(H)}.$ 0 and minimum degree at least $t_H(G)\ge p^{e(H)}.$ 1, and is neither a cycle nor complete bipartite, then for signed graphons $t_H(G)\ge p^{e(H)}.$ 2,

$t_H(G)\ge p^{e(H)}.$ 3

which is then used to control the signed expansion of $t_H(G)\ge p^{e(H)}.$ 4 near the constant graphon (Lovász, 2010).

A further analytic reduction uses symmetric hosts. Szegedy showed that it suffices to verify Sidorenko’s inequality on highly symmetric Cayley-type hosts; in particular, one may reduce to vertex-transitive and edge-transitive settings, and then to Cayley graphs (Zhao, 21 Jul 2025). More recently, a group-theoretic refinement showed that if, for a fixed bipartite graph $t_H(G)\ge p^{e(H)}.$ 5, the $t_H(G)\ge p^{e(H)}.$ 6-density of every Cayley-type host is at least the $t_H(G)\ge p^{e(H)}.$ 7-density of its conjugacy-class average, then $t_H(G)\ge p^{e(H)}.$ 8 is strong Sidorenko (Zhao, 13 Jun 2026). This reduction is conditional, but it organizes the problem around representation-theoretic positivity.

5. Product constructions, subdivisions, and recent expansions of the Sidorenko class

A major source of new examples comes from closure results. If $t_H(G)\ge p^{e(H)}.$ 9 is a tree and $H$ 0 has Sidorenko’s property, then the Cartesian product $H$ 1 also has Sidorenko’s property (Kim et al., 2013). This implies that all $H$ 2-dimensional grids with arbitrary side lengths satisfy Sidorenko’s conjecture (Kim et al., 2013). A later result proved an analogous closure under product with even cycles: if $H$ 3 is Sidorenko, then $H$ 4 is Sidorenko for every $H$ 5 (Conlon et al., 2015).

Subdivisions form another robust family. Conlon, Kim, Lee, and Lee proved that subdivisions of certain graphs, including cliques, have Sidorenko’s property via locally dense counting and bounded-degree reduction (Conlon et al., 2015). Conlon and Lee later showed that for every bipartite graph $H$ 6 with bipartition $H$ 7, there exists an integer $H$ 8 such that the blow-up $H$ 9 obtained by gluing $H$ 0 copies along $H$ 1 is Sidorenko; in particular, $H$ 2 suffices (Conlon et al., 2018). Equivalently, every bipartite $H$ 3 satisfies an $H$ 4-version of Sidorenko’s conjecture for some $H$ 5 (Conlon et al., 2018).

Recent work sharpens the substitution picture. Replacing each edge of a graph by an even generalized theta graph yields a Sidorenko graph whenever the base graph satisfies the KNRS conjecture, and in particular replacing each edge of a complete graph by an even generalized theta graph is unconditionally Sidorenko (Im et al., 2024). The same paper proves Sidorenko for broad non-uniform edge replacements by even paths under a divisibility condition on the total numbers of paths of each length (Im et al., 2024).

The abelian-Cayley-host setting gives an orthogonal generalization. Every even subdivision of an arbitrary graph satisfies Sidorenko’s inequality when the host is a Cayley graph over a finite abelian group (Zhao, 21 Jul 2025). The proof encodes cycle constraints by a circuit matrix and uses a Fourier expansion over $H$ 6; even lengths force Fourier coefficients to pair as nonnegative squares, so the principal character alone yields the lower bound (Zhao, 21 Jul 2025). In the abelian case, this recovers such examples as $H$ 7 viewed as an even subdivision of $H$ 8.

A recent non-abelian variant proves a Sidorenko-type inequality for 1-subdivision graphs on conjugacy-averaged Cayley kernels associated with arbitrary real-valued functions on finite groups (Zhao, 13 Jun 2026). Since even subdivisions can be realized as 1-subdivisions of other graphs, this gives a broad class of Sidorenko-type inequalities in conjugacy-averaged non-abelian settings (Zhao, 13 Jun 2026).

Sidorenko’s conjecture has strong connections to quasirandomness. A bipartite graph $H$ 9 is forcing if $H=(A\cup B,E(H))$ 0 and $H=(A\cup B,E(H))$ 1 together imply that $H=(A\cup B,E(H))$ 2 is quasirandom. Graphs with two universal vertices on one side are forcing (Conlon et al., 2010), and Li–Szegedy proved the forcing conjecture for the universal-vertex class using logarithmic calculus (Li et al., 2011). This relation explains why the conjecture is often viewed as a lower-bound counterpart to quasirandomness characterization.

There is also a determinant and entropy viewpoint through homogeneous Gaussian Markov random fields. If $H=(A\cup B,E(H))$ 3 is bipartite, then the differential entropy of any homogeneous GMRF on $H=(A\cup B,E(H))$ 4 is at least $H=(A\cup B,E(H))$ 5 times the edge entropy plus $H=(A\cup B,E(H))$ 6 times the point entropy (Csikvári et al., 2018). In determinant language this becomes

$H=(A\cup B,E(H))$ 7

where $H=(A\cup B,E(H))$ 8 is the maximum determinant of a covariance matrix with diagonal $H=(A\cup B,E(H))$ 9 and edge correlations $H$ 00 (Csikvári et al., 2018). This inequality follows from Sidorenko for $H$ 01 via a large deviation principle on high-dimensional spheres, but some determinant inequalities can also be proved for graphs not yet known to satisfy Sidorenko’s conjecture (Csikvári et al., 2018).

Another viewpoint concerns sums of squares. In the gluing-algebra framework, a Sidorenko inequality for $H$ 02 would amount to nonnegativity of $H$ 03. However, some true Sidorenko inequalities cannot be certified by sums of squares. In particular, if $H$ 04 is a trivial square, then $H$ 05 has no sums-of-squares certificate (Garg et al., 2022). Computational work classified bipartite graphs on at most seven edges according to whether such a certificate exists, showing that the SoS obstruction goes beyond trivial squares (Garg et al., 2022).

Open problems remain abundant. The extension of the Cartesian product theorem beyond trees is explicitly identified as nontrivial because the current proof requires usable upper bounds on $H$ 06 when a negative exponent appears (Kim et al., 2013). The operation $H$ 07 on non-bipartite graphs, which reduces to $H$ 08 in the bipartite case, leads to concrete questions such as whether $H$ 09 can have Sidorenko’s property; already $H$ 10 is highlighted as a minimal unknown case in the older literature (Kim et al., 2013). Recent symmetry-based approaches pose a different problem: to prove that conjugacy averaging cannot decrease $H$ 11-density for broad classes of Cayley kernels, which would imply strong Sidorenko via reduction (Zhao, 13 Jun 2026).

Taken together, these results suggest a conjectural picture in which Sidorenko’s property is stable under far more recursive decompositions, substitutions, and symmetry reductions than are currently proved. What remains missing is a general principle broad enough to subsume tree-arrangeability, higher tree decompositions, thick-graph constructions, biregular reductions, theta substitutions, and Cayley-host positivity within a single theorem.