Sidorenko’s conjecture for subdivisions and theta substitutions

One fundamental problem in extremal graph theory is to determine the maximum/minimum possible number of $H$ copies in graphs of given edge density. A major conjecture by Sidorenko [19, 18] and independently Erdős and Simonovits [9] states that for every bipartite graph $H$, this number is asymptotically minimized by the random graph of the same edge density. Formally, for graphs $H$ and $G$, a homomorphism from $H$ to $G$ is a function $f:V(H)\rightarrow V(G)$ such that $f(u)f(v)\in E(G)$ whenever $uv\in E(H)$. Let $hom(H,G)$ be the number of homomorphisms from $H$ to $G$. The homomorphism density of $H$ in $G$, denoted by $t_{H}(G)$, is defined by $\frac{hom(H,G)}{v(G)^{v(H)}}$. If $v(G)$ is sufficiently large compared to $v(H)$, then the homomorphism density is asymptotically the same as the subgraph density. Sidorenko’s conjecture is stated as follows.

We say a graph $H$ is Sidorenko if (1) holds for every graph $G$. Sidorenko [18] confirmed the conjecture for trees, even cycles, complete bipartite graphs, and when one of the parts of the bipartition has size at most $4$. Despite attracting continuous interests and efforts, and many partial results over recent years (see [4, 5, 6, 7, 10, 11, 15, 20]), Sidorenko’s conjecture is still far from being completely understood.

Note that the bipartiteness condition in Sidorenko’s conjecture is necessary since $G$ is possibly a bipartite graph. One may ask which condition of $G$ ensures that the (asymptotically) same inequality holds for a nonbipartite graph $H$. Chung, Graham, and Wilson [3] proved a celebrity result on the characterization of quasirandomness which implies that if for every $U\subseteq V(G)$, the number of edges inside $U$ is $d{|U|\choose 2}+o(v(G)^{2})$, then $t_{H}(G)=d^{e(H)}+o(1)$. Kohayakawa, Nagle, Rödl, and Schacht [12] conjectured that one-side inequality is sufficient to guarantee the lower bound, which is now called KNRS conjecture. A graph $G$ on $n$-vertex is called $(\varepsilon,d)$-dense if for every $U\subseteq V(G)$ with $|U|\geq\varepsilon n$, there holds $e(G[U])\geq d{|U|\choose 2}$. Informally, such a graph $G$ is called locally dense.

We say $H$ is KNRS if $H$ satisfies the KNRS conjecture. While some partial cases such as complete graphs, complete multipartite graphs [12], odd cycles [17], see also [1, 13, 14] for recent developments, the full conjecture is still widely open.

It is clear that if $H$ is Sidorenko, then it is KNRS. Conlon, Kim, Lee, and Lee [5] showed more connections between these two conjectures: if $H$ is KNRS, then its $1$-subdivision is Sidorenko. Furthermore, they proved that if one replaces each edge of a KNRS graph $H$ by $K_{2,t}$, then the resulting graph is Sidorenko. A graph $H$ is called a generalized theta graph if it is obtained by adding internally disjoint paths between two distinct vertices $u$ and $v$. We call $u$ and $v$ the roots of $H$. If all these paths have even number of edges, then we call $H$ an even generalized theta graph. Note that $K_{2,t}$ is the simplest even generalized theta graph. Conlon, Kim, Lee and Lee [5] also proved that an even generalized theta graph is Sidorenko.

Our first result is a common generalization of these two results, providing a new class of Sidorenko graphs.

Very recently, Chen, Lin, and Ma [2] independently proved that if $H$ is KNRS, then its $(2k-1)$-subdivision is Sidorenko. While some ideas of our proof and Chen, Lin, and Ma’s are close: defining auxiliary weighted graphs that count the number of paths between two vertices, and showing that if $G$ is almost regular, then these weighted graphs are locally dense. However, to extend paths to generalized theta graphs, we need to show that the Hadamard product of those auxiliary graphs is also locally dense. As the locally dense property is not preserved by the Hadamard product, we need to prove that those auxiliary graphs are correlated.

The “subdivision results” and the “$\Theta$ results” aforementioned are processing a uniform replacement of edges of $H$, and more importantly they all require $H$ to be KNRS. Our next result goes beyond in both aspects, showing that for any graph $H$, the graph obtained from replacing edges of $H$ by any non-uniform even generalized theta graphs is Sidorenko provided that the number of paths satisfies a certain divisibility condition.

The proof of Theorem 1.4 builds on Theorem 1.3 and uses a Hölder trick inspired by Conlon and Lee [6]. Conlon and Lee [6] proved that for every bipartite graph $H$, there exists a bipartite graph $H^{\prime}$ such that their disjoint union $H\cup H^{\prime}$ is Sidorenko. Using Theorem 1.4, we have an analogous statement that if $H$ is a subdivision of a graph, then $H^{\prime}$ can be taken as a generalized theta graph.

We remark that a result of a similar spirit for KNRS conjecture was recently proved by Kráľ, Volec and Wei [13], who showed that for every graph $H$ with girth at least $50$, there exists a larger graph $H’$, which is KNRS and containing $H$ as an induced subgraph.

While Theorem 1.4 gives a new approach to consider non-uniform replacement, we are not able to deal with subdivisions with different path lengths. In the consideration of non-uniform subdivisions, we prove a special case of clique subdivision is Sidorenko.

Theorem 1.6 is in fact a special case of a more general theorem. To state it, we need a notion of product of two graphs introduced by Bradač, Sudakov and Wigderson [1]. For graphs $H_{1},H_{2}$, an independent set $I\subseteq V(H_{1})$, and a vertex $a\in V(H_{1})\setminus I$, let $H_{1}\ltimes_{I}^{a}H_{2}$ be a graph obtained by the following process. We start with $|V(H_{2})|$ copies of $H_{1}$ that are identified at $I$ and disjoint otherwise. Let $X$ be the collection of copies of the vertex $a$ from each $H_{1}$. We add a copy of $H_{2}$ on $X$ to obtain $H_{1}\ltimes_{I}^{a}H_{2}$. Bradač, Sudakov and Wigderson [1] recently proved that if $H_{1}$ and $H_{2}$ are KNRS, then $H_{1}\ltimes_{I}^{a}H_{2}$ is also KNRS. We note that the product $H_{1}\ltimes_{I}^{a}H_{2}$ may not be bipartite even if $H_{1}$ and $H_{2}$ are Sidorenko (considering the case that $H_{1}$ is a path and $H_{2}$ is an edge). So a direct analog of Bradač, Sudakov and Wigderson’s result is not true for Sidorenko’s conjecture. However, our next result shows that if we subdivide the product (minimally) to make it bipartite, then it becomes Sidorenko.

We note that our proof provides a stronger statement that instead of using a subdivision of edges corresponding to $H_{2}$, one can replace those edges with a theta graph of even lengths.

Recall that even generalized theta graphs are Sidorenko. It is still not known whether a generalized theta graph is KNRS; Bradač, Sudakov, and Wigderson [1] proved that it satisfies a slightly weaker “regular KNRS conjecture”. We prove that if we identify two roots of a generalized theta graph, then it is KNRS. We call such graphs flowers. Equivalently, a flower is a graph consisting of a collection of cycles sharing a common vertex and disjoint otherwise.

By considering the $1$-subdivision of a flower, we see that a flower consisting of even cycles satisfies Sidorenko’s conjecture, which recovers a result in [5].

We will work with graphons instead of graphs. A graphon is a measurable function $W:[0,1]^{2}\rightarrow[0,1]$ such that $W(x,y)=W(y,x)$ for (almost) every $x,y\in[0,1]$. A graphon $W$ is $d$-regular if $\int_{[0,1]}W(x,y)\mathrm{d}y=d$ for (almost) every $x\in[0,1]$. Graphons are considered as a limit object of dense graphs and many statements of graphs have an equivalent form in terms of graphon. See [16] for the theory of graphon and examples.

For a graph $H$ and a graphon $W$, the homomorphism density of $H$ in $W$ is defined as

With this notation, Sidorenko’s conjecture can be stated in the following equivalent form.

In the graph version of Sidorenko’s conjecture, one may assume that the host graph $G$ is regular [21]. An analogous statement for the graphon version is also true.

We can also consider KNRS conjecture in terms of graphon. We first define a notion that corresponds to locally denseness. Let $\lambda$ be the standard Lebesgue measure on $[0,1]$.

The graphon version of KNRS conjecture reads as follows.

Reiher [17] proved the following lemma which is useful for KNRS conjecture. It says that if a graph is locally dense, then it is also locally dense in a weighted sense.

We will use the following graphon version.

We also note the following extension of Reiher’s lemma proved by Bradač, Sudakov, and Wigderson [1]. Roughly speaking, the original Reiher’s lemma is for vertex-weighted counting of $K_{2}$ and it extends to vertex-weighted counting of a KNRS graph $H$.

We need the following version of Hölder’s inequality.