1. Introduction
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 [Reference Sidorenko1,Reference Sidorenko2] (conjectured earlier in a weaker form by Erdős and Simonovits [Reference Erdős and Simonovits3]) states that for every bipartite graph
$H$
, this number is asymptotically minimised 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.
Conjecture 1.1 (Sidorenko’s conjecture). For every bipartite graph
$H$
and every graph
$G$
, we have
\begin{equation} t_H(G) \geq t_{K_2}(G)^{e(H)}. \end{equation}
We say a graph
$H$
is Sidorenko if (1) holds for every graph
$G$
. Sidorenko [Reference Sidorenko1] 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 interest and effort and many partial results over recent years (see [Reference Conlon, Fox and Sudakov4–Reference Szegedy11]), 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 same (asymptotic) inequality holds for a nonbipartite graph
$H$
. Chung, Graham, and Wilson [Reference Chung, Graham and Wilson12] proved a celebrated result on the characterisation of quasirandomness which implies that if for every
$U \subseteq V(G)$
, the number of edges inside
$U$
is
$d \binom{|U|}{2} + o(v(G)^2)$
, then
$t_H(G) = d^{e(H)}+o(1)$
. Kohayakawa, Nagle, Rödl, and Schacht [Reference Kohayakawa, Nagle, Rödl and Schacht13] conjectured that one-sided 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
$e(G[U]) \geq d \binom{|U|}{2}$
. Informally, such a graph
$G$
is called locally dense.
Conjecture 1.2 (KNRS conjecture). Let
$H$
be a graph. Then for every
$d, \eta \in (0, 1)$
, there exists
$\varepsilon = \varepsilon (d, \eta , H)$
such that if
$G$
is
$(\varepsilon , d)$
-dense, then
$t_H(G) \geq (1-\eta )d^{e(H)}$
.
We say
$H$
is KNRS if
$H$
satisfies the KNRS conjecture. While some partial cases such as complete graphs, complete multipartite graphs [Reference Kohayakawa, Nagle, Rödl and Schacht13], and odd cycles [Reference Reiher14], see also [Reference Bradač, Sudakov and Wigderson15–Reference Lee17] for recent developments, the full conjecture is still wide open.
It is clear that if
$H$
is Sidorenko, then it is KNRS. Conlon, Kim, Lee, and Lee [Reference Conlon, Kim, Lee and Lee5] 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 generalised 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 generalised theta graph. Note that
$K_{2,t}$
is the simplest even generalised theta graph. Conlon, Kim, Lee and Lee [Reference Conlon, Kim, Lee and Lee5] also proved that an even generalised theta graph is Sidorenko.
Our first result is a common generalisation of these two results, providing a new class of Sidorenko graphs.
Theorem 1.3.
Let
$H$
be a graph that satisfies the KNRS conjecture. Let
$\Theta$
be an even generalised theta graph with roots
$u$
and
$v$
. If we use
$\Theta$
to replace each edge of
$H$
by identifying its two vertices with
$u$
and
$v$
respectively, then the resulting graph is Sidorenko. In particular, every even generalised theta graph is Sidorenko.
Very recently, Chen, Lin, and Ma [Reference Chen, Lin and Ma18] independently proved that if
$H$
is KNRS, then its
$(2k-1)$
-subdivision is Sidorenko. 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 generalised 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
$H$
(i.e., replacing each edge with the same even generalised theta graph), 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 generalised theta graphs is Sidorenko provided that the number of paths satisfies a certain divisibility condition.
Theorem 1.4.
Let
$H$
be an
$h$
-vertex graph and
$H'$
be a graph obtained by replacing each edge
$e$
of
$H$
with internally vertex-disjoint paths of even lengths connecting the end vertices of
$e$
. Let
$h_e(k)$
denote the number of length
$k$
paths in the replacement of
$e$
. If
$\sum _{e \in E(H)} h_e(2k)$
is divisible by
$\binom{h}{2}$
for all
$k \geq 1$
, then
$H'$
satisfies Sidorenko’s conjecture. Also, if
$\sum _{e \in E(H)} h_e(2k)=0$
for all but one
$k$
and
$\sum _{e \in E(H)} h_e(2k) \geq {\binom{h}{2}}$
for this exceptional
$k$
, then
$H'$
is Sidorenko.
The proof of Theorem 1.4 builds on Theorem 1.3 and uses a Hölder trick inspired by the work of Conlon and Lee [Reference Conlon and Lee6], who proved that for every bipartite graph
$H$
, there exists a bipartite graph
$H'$
such that their disjoint union
$H \cup H'$
is Sidorenko. Using Theorem 1.4, we have an analogous statement that if
$H$
is a subdivision of a graph, then
$H'$
can be taken as a generalised theta graph.
Corollary 1.5.
Let
$H_0$
be a graph and
$H$
be a subdivision of
$H_0$
such that every edge is subdivided an odd number of times. Then there exists a generalised theta graph
$\Theta$
such that
$H \cup \Theta$
is Sidorenko.
Proof sketch. Let
$h(2k)$
be the number of edges of
$H_0$
subdivided
$2k-1$
times in
$H$
, and let
$K \subseteq \mathbb{Z}_{\geq 0}$
be the set containing all
$k$
with
$h(2k) \neq 0$
. We take
$\Theta$
to be a generalised even theta graph such that it is obtained by adding
$\binom {v(H)+2}{2}-h(2k)$
many paths of length
$2k$
between two fixed vertices for every
$k \in K$
. Then we can view
$H \cup \Theta$
as a graph obtained by replacing each edge of
$H \cup K_2$
with an even generalised theta graph. So by Theorem 1.4,
$H \cup \Theta$
is Sidorenko.
We remark that a result of a similar spirit for the KNRS conjecture was recently proved by Kráľ, Volec and Wei [Reference Král͏̌, Volec and Wei16], who showed that for every graph
$H$
with girth at least
$50$
, there exists a larger graph
$H'$
, which is KNRS and contains
$H$
as an induced subgraph.
While Theorem 1.4 gives a new approach to consider non-uniform replacement (i.e., each edge of
$H$
is replaced by a different graph), we are not able to deal with subdivisions with different path lengths. Indeed, to guarantee
$\sum _{e \in E(H)} h_e(2k) \geq \binom {h}{2}$
, at least one edge is replaced by a generalised theta graph with at least two paths except when
$H'$
is a
$(2k-1)$
-subdivision of
$K_h$
. In the consideration of non-uniform subdivisions, we prove that a special case of clique subdivision is Sidorenko.
Theorem 1.6.
Let
$h, \ell _1, \ell _2 \geq 1$
be integers and
$v \in V(K_h)$
be a vertex. Let
$H$
be a subdivision of
$K_h$
obtained by subdividing each edge
$2\ell _1-1$
times if an edge is not incident to
$v$
and
$\ell _2$
times if an edge is incident to
$v$
. Then
$H$
is Sidorenko.
Theorem 1.6 is in fact a special case of a more general theorem. To state it, we need a notion for the product of two graphs introduced by Bradač, Sudakov and Wigderson [Reference Bradač, Sudakov and Wigderson15]. 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 [Reference Bradač, Sudakov and Wigderson15] 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 analogue 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.
Theorem 1.7.
Let
$H_1$
be a Sidorenko graph and
$H_2$
be a KNRS graph. Let
$I \subseteq V(H_1)$
be an independent set and
$a \in V(H_1) \setminus I$
be a vertex and
$k \geq 1$
be an integer. Let
$F$
be a graph obtained by subdividing
$2k-1$
times edges of
$H_1 \ltimes _I^a H_2$
that correspond to
$H_2$
. Then
$F$
is 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.
Proof of Theorem 1.6. We take
$H_1$
to be a path of length
$\ell _2+1$
and
$H_2 = K_{h-1}$
. The set
$I$
consists of one of the end vertices of
$H_1$
and
$a$
is the other end vertex of
$H_1$
. Then subdividing of
$H_1 \ltimes _I^a H_2$
that corresponds to
$H_2$
by
$2\ell _1-1$
times produces the desired graph.
Recall that even generalised theta graphs are Sidorenko. It is still not known whether a generalised theta graph is KNRS; Bradač, Sudakov, and Wigderson [Reference Bradač, Sudakov and Wigderson15] proved that it satisfies a slightly weaker ‘regular KNRS conjecture’. We prove that if we identify the two roots of a generalised 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.
Theorem 1.8. Any flower is KNRS.
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 [Reference Conlon, Kim, Lee and Lee5].
Organisation. In Section 2 we introduce the notion of graphon and related lemmas. In Section 3.1, we first show that certain auxiliary graphs are locally dense and prove Theorems 1.3 and 1.7. We prove Theorem 1.4 in Section 3.2 and Theorem 1.8 in Section 3.3. Concluding remarks are given in Section 4.
2. Preliminaries
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 graphons. See [Reference Lovász19] for a gentle introduction to the theory of graphons and numerous illustrative examples.
For a graph
$H$
and a graphon
$W$
, the homomorphism density of
$H$
in
$W$
is defined as
\begin{equation*}t_H(W) = \int _{[0, 1]^{v(H)}} \prod _{ij \in E(H)} W(x_i, x_j) \prod _{i \in [v(H)]} \mathrm{d}x_i.\end{equation*}
With this notation, Sidorenko’s conjecture can be stated in the following equivalent form.
Conjecture 2.1.
For every bipartite graph
$H$
and graphon
$W$
, we have
$t_H(W) \geq t_{K_2}(W)^{e(H)}$
.
In the graph version of Sidorenko’s conjecture, one may assume that the host graph
$G$
is regular [Reference Szegedy20]. An analogous statement for the graphon version is also true.
Lemma 2.2 ([Reference Coregliano and Razborov21], Theorem 8.2). A bipartite graph
$H$
is Sidorenko if and only if
$t_H(W) \geq d^{e(H)}$
for every
$d$
-regular graphon
$W$
.
We can also consider the KNRS conjecture in terms of graphons. We first define a notion that corresponds to locally denseness. Let
$\lambda$
be the standard Lebesgue measure on
$[0, 1]$
.
Definition 2.3.
A graphon
$W$
is
$d$
-locally dense if for every measurable subset
$S \subseteq [0, 1]$
, the following holds:
\begin{equation*}\int _{S \times S} W(x, y) \mathrm{d}x\mathrm{d}y \geq d \lambda (S)^2.\end{equation*}
The graphon version of the KNRS conjecture reads as follows.
Lemma 2.4 ([Reference Bradač, Sudakov and Wigderson15], Lemma 2.6). A graph
$H$
is KNRS if and only if
$t_H(W) \geq d^{e(H)}$
for every
$d$
-locally dense graphon
$W$
.
Reiher [Reference Reiher14] proved the following lemma which is useful for the KNRS conjecture. It says that if a graph is locally dense, then it is also locally dense in a weighted sense.
Lemma 2.5 ([Reference Reiher14]). Let
$G$
be an
$n$
-vertex
$(\varepsilon , d)$
-dense graph. Let
$f\,:\,V(G) \rightarrow [0, 1]$
be a function such that
$\sum _{v \in V(G)} f(v) \geq \varepsilon n$
. Then
$\sum _{uv \in E(G)} f(u)f(v) \geq d \left (\sum _{v \in V(G)} f(v)\right )^2-n.$
We will use the following graphon version.
Lemma 2.6 ([Reference Bradač, Sudakov and Wigderson15], Lemma 2.8). Let
$W$
be a
$d$
-locally dense graphon and
$f\,:\,[0, 1] \to [0, 1]$
be a measurable function. Then,
\begin{equation*}\int _{[0, 1]^2} f(x)f(y)W(x, y)\mathrm{d}x\mathrm{d}y \geq d\left (\int _{[0, 1]} f(x)\mathrm{d}x\right )^2.\end{equation*}
We also note the following extension of Reiher’s lemma proved by Bradač, Sudakov, and Wigderson [Reference Bradač, Sudakov and Wigderson15]. 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
$H$
.
Lemma 2.7 ([Reference Bradač, Sudakov and Wigderson15], Lemma 2.10). Let
$H$
be a KNRS graph and
$W$
be a
$d$
-locally dense graphon. Let
$f\,:\,[0, 1] \to [0, 1]$
be a measurable function. Then,
\begin{equation*}\int _{[0, 1]^{v(H)}} \prod _{i \in [v(H)]} f(x_i) \prod _{ij \in E(H)} W(x_i, x_j) \prod _{i \in [v(H)]} \mathrm{d}x_i \geq d^{e(H)}\left (\int _{[0, 1]} f(x)\mathrm{d}x\right )^{v(H)}.\end{equation*}
We need the following version of Hölder’s inequality.
Theorem 2.8 (Hölder’s inequality). Let
$p_1, \ldots , p_k, r \in (0, \infty )$
satisfies
$\frac {1}{p_1} + \ldots + \frac {1}{p_k} = \frac {1}{r}$
. Then for any measurable functions
$f_1, \ldots , f_k\,:\,[0, 1] \rightarrow \mathbb{R}$
, we have
\begin{equation*}\prod _{i=1}^k \left (\int _{[0, 1]}|f_i|^{p_i} \mathrm{d}x\right )^{1/p_i} \geq \left (\int _{[0, 1]} \left (\prod _{i=1}^k |f_i| \right )^{r}\mathrm{d}x\right )^{1/r}.\end{equation*}
2.1 Outline of proofs
We outline the proofs of our main results. For Theorem 1.3, we first define an auxiliary graphon
$W^\Theta$
by mapping each pair
$(x, y)$
to the density of homomorphisms of
$\Theta$
in
$W$
where the two end vertices are mapped to
$x$
and
$y$
. Our main lemma (Lemma 3.2) states that if
$W$
is
$d$
-regular, then
$W^{\Theta }$
is
$d^{e(\Theta )}$
-locally dense. Theorem 1.3 is then a direct consequence of this lemma. The proof of Theorem 1.7 is similar, but it additionally requires us to consider vertex weights with respect to the weighted counts of
$H_1$
.
To prove Theorem 1.4, which deals with non-uniform substitution, we apply Hölder’s inequality to all permutations of
$V(H)$
. We can then reduce the non-uniform case, under an additional condition, to the uniform case covered by Theorem 1.3.
Finally, to prove Theorem 1.8, we apply Reiher’s lemma to each of the odd cycles. This reduces the flower counting problem to a tree counting problem. As trees are Sidorenko, we can infer that flowers are KNRS.
3. Proofs of main results
3.1 Locally dense auxiliary graph
For a graphon
$W$
and a graph
$F$
with vertex set
$[f]$
and two (distinct) roots
$i,j \in [f]$
, we define the
$F$
-counting kernel
$W^F$
as
\begin{equation*}W^F(x_i, x_j)\,:=\,\int _{[0, 1]^{f-2}} \prod _{i_1i_2 \in E(F)} W(x_{i_1}, x_{i_2}) \prod _{k \in [f] \setminus \{i, j\}} \mathrm{d}x_k.\end{equation*}
We note that if
$F$
is not symmetric, then
$W^F$
may not be a graphon. To avoid possible technicalities from this issue, in this paper, we assume that
$F$
has an automorphism that swaps the two roots. Indeed, this assumption ensures that
$W^F$
is a graphon. When
$F$
is a path of length
$k$
with two roots being its end vertices, we denote
$W^F$
by
$W^k$
. Note that
$W^k$
is equivalent to the
$k$
-th matrix power of
$W$
. The following property of
$W^F$
is useful.
Lemma 3.1.
Let
$F$
be a graph with two roots such that there exists an automorphism that swaps its two roots and
$H$
be a graph. Let
$H'$
be a graph obtained by replacing each edge
$e$
of
$H$
with a copy of
$F$
such that the roots of
$F$
are identified with the end vertices of
$e$
. Then
$t_{H'}(W) = t_H(W^F)$
for every graphon
$W$
.
As it follows from writing the definition of
$t_{H'}(W)$
and integrating variables corresponding to the vertices of
$V(H') \setminus V(H)$
first, we omit the proof of Lemma 3.1.
The key lemma of this subsection is the following.
Lemma 3.2.
Let
$W$
be a
$d$
-regular graphon. Let
$\Theta$
be an even generalised theta graph. Then
$W^{\Theta }$
is
$d^{e(\Theta )}$
-locally dense.
We now prove this lemma by inductively attaching even paths to the roots of an even generalised theta graph. The next lemma captures the essence of this inductive process.
Lemma 3.3.
Let
$W_1$
be a
$d_1$
-locally dense graphon and
$W_2$
be a
$d_2$
-regular graphon and
$k \geq 1$
be an integer. Define a graphon
$W$
with
$W(x, y) = W_1(x, y)W_2^{2k}(x, y)$
for all
$x,y\in [0,1]$
. Then
$W$
is
$d_1d_2^{2k}$
-locally dense.
Proof. We first note that as
$W_2$
is
$d_2$
-regular, we have
\begin{align*} \int _{[0, 1]} W_2^{k}(x, y) \mathrm{d}y & = \int _{[0, 1]^{k}} W_2(x, y_1)W_2(y_1, y_2) \cdots W_2(y_{k-1}, y_k) \prod _{i \in [k]} \mathrm{d}y_i \\ & = \int _{[0, 1]^{k-1}}\! W_2(x, y_1)W_2(y_1, y_2) \cdots W_2(y_{k-2}, y_{k-1})\! \int _{[0, 1]}\! W_2(y_{k-1}, y_k) \mathrm{d}y_k\! \prod _{i \in [k-1]}\! \mathrm{d}y_i \\ & = d_2 \int _{[0, 1]^{k-1}} W_2(x, y_1)W_2(y_1, y_2) \cdots W_2(y_{k-2}, y_{k-1}) \prod _{i \in [k-1]} \mathrm{d}y_i. \end{align*}
Therefore, by induction, we have
$\int _{[0, 1]} W_2^{k}(x, y) \mathrm{d}y = d_2^k$
for every
$k \geq 1$
and almost every
$x \in [0, 1]$
. Let
$S \subseteq [0, 1]$
be any measurable set with positive measure. Then we have
\begin{align*} \int _{S \times S} W(x, y) \mathrm{d}x\mathrm{d}y &= \int _{S \times S} W_1(x, y)W_2^{2k}(x, y) \mathrm{d}x\mathrm{d}y \\ &= \int _{S \times S} W_1(x, y) \int _{[0, 1]} W_2^k(z, x)W_2^k(z, y) \mathrm{d}z\mathrm{d}x\mathrm{d}y \\ & = \int _{[0, 1]}\int _{S \times S} W_2^k(z, x)W_1(x, y)W_2^k(z, y) \mathrm{d}x\mathrm{d}y\mathrm{d}z \\ & \geq \int _{[0, 1]} d_1 \left (\int _S W_2^k(z, x)\mathrm{d}x\right )^2 \mathrm{d}z \end{align*}
by applying Lemma 2.6 with
$f( \cdot ) = W_2^k(z, \cdot ) \cdot \mathbf{1}_{S}$
. By using Jensen’s inequality for
$x \mapsto x^2$
, we have
\begin{align*} d_1 \int _{[0, 1]} \left (\int _S W_2^k(z, x)\mathrm{d}x\right )^2 \mathrm{d}z & \geq d_1\left (\int _{[0, 1]} \int _S W_2^k(z, x) \mathrm{d}x \mathrm{d}z \right )^2 \\ & = d_1 \left (\int _{S} \int _{[0, 1]} W_2^k(z, x) \mathrm{d}z \mathrm{d}x \right )^2 \\ & = d_1 d_2^{2k}\lambda (S)^2. \end{align*}
Therefore,
$\int _{S \times S} W(x, y) \mathrm{d}x\mathrm{d}y \geq d_1d_2^{2k}\lambda (S)^2$
for any measurable set
$S \subseteq [0, 1]$
which concludes the proof.
We now prove Lemma 3.2.
Proof of Lemma 3.2. We use induction on the number of paths in
$\Theta$
. If
$\Theta$
consists of a single path of length
$2k$
, then we apply Lemma 3.3 with
$W_1 \equiv 1$
and
$W_2=W$
. Then
$W^{\Theta } = W^k$
is
$d^{2k}$
-locally dense by Lemma 3.3.
If
$\Theta$
has more than one path, then let
$\Theta '$
be a theta graph obtained by deleting one path of length
$2k$
from
$\Theta$
. By the induction hypothesis,
$W^{\Theta '}$
is
$d^{e(\Theta ')}$
-locally dense. We apply Lemma 3.3 with
$W_1=W^{\Theta '}$
and
$W_2=W$
. Then the Hadamard product of
$W^{\Theta '}$
and
$W^{2k}$
is
$d^{e(\Theta ')}d^{2k}=d^{e(\Theta )}$
-locally dense by Lemma 3.3. As the Hadamard product of
$W^{\Theta '}$
and
$W^{2k}$
is
$W^{\Theta }$
, it concludes the proof.
We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3. Let
$H$
be a KNRS graph and
$\Theta$
be an even generalised theta graph. Let
$H'$
be a graph obtained by replacing each edge of
$H$
by
$\Theta$
. Let
$W$
be a
$d$
-regular graphon. Then by Lemma 3.2,
$W^{\Theta }$
is
$d^{e(\Theta )}$
-locally dense. Therefore,
$t_{H'}(W) = t_{H}(W^{\Theta }) \geq d^{e(\Theta )e(H)} = d^{e(H')}$
as
$H$
is KNRS. Therefore,
$H'$
is Sidorenko.
We now prove Theorem 1.7.
Proof of Theorem 1.7. Let
$W$
be a
$d$
-regular graphon. We label the vertices of
$H_1$
by
$1, 2, \ldots , h$
where
$I=\{1, 2, \ldots , t\}$
and
$a=h$
. For each
$x_1, \ldots , x_t \in [0, 1]$
, we define the normalised number of embeddings of
$H_1$
(with
$1, 2, \ldots , t,$
and
$h$
being fixed) by
\begin{equation*}f_{x_1, \ldots , x_t}^W(x_h) = \int _{[0, 1]^{h-t-1}} \prod _{ij \in E(H_1)} W(x_i, x_j) \prod _{t+1 \leq i \leq h-1} \mathrm{d}x_i.\end{equation*}
Then we observe that
\begin{equation*}t_{F}(W) = \int _{[0, 1]^{t}} \left ( \int _{[0, 1]^{v(H_2)}} \prod _{i \in [v(H_2)]} f_{x_1, \ldots , x_t}^W(y_i) \prod _{ij \in E(H_2)} W^{2k}(y_i, y_j) \prod _{i \in [v(H_2)]} \mathrm{d}y_i\right ) \prod _{i \in [t]} \mathrm{d}x_i.\end{equation*}
As
$W^{2k}$
is
$d^{2k}$
-locally dense by Lemma 3.2 and
$H_2$
is KNRS, we can apply Lemma 2.7 to obtain
\begin{equation*}t_{F}(W) \geq \int _{[0, 1]^{t}} \left ( \int _{[0, 1]} f_{x_1, \ldots , x_t}^W(z) \mathrm{d}z \right )^{v(H_2)} d^{2ke(H_2)} \prod _{i \in [t]} \mathrm{d}x_i.\end{equation*}
By Jensen’s inequality for
$x \mapsto x^{v(H_2)}$
and that
$H_1$
is Sidorenko, we have
\begin{align*} t_{F}(W) &\geq d^{2ke(H_2)} \int _{[0, 1]^{t}} \left ( \int _{[0, 1]} f_{x_1, \ldots , x_t}^W(z) \mathrm{d}z \right )^{v(H_2)} \prod _{i \in [t]} \mathrm{d}x_i \\ & \geq d^{2ke(H_2)} \left (\int _{[0, 1]^{t}} \int _{[0, 1]} f_{x_1, \ldots , x_t}^W(z) \mathrm{d}z \prod _{i \in [t]} \mathrm{d}x_i \right )^{v(H_2)} \\ & \geq d^{2ke(H_2) + v(H_2)e(H_1)}. \end{align*}
Therefore,
$F$
is Sidorenko.
3.2 Uniformization via Hölder’s inequality
This section’s main lemma is the following, which allows us to reduce a non-uniform replacement case into a uniform case.
Lemma 3.4.
Let
$H$
be a graph with
$V(H)=[h]$
and
$H'$
be a graph obtained by replacing each edge
$e$
of
$H$
with internally vertex-disjoint paths of even lengths connecting the end vertices of
$e$
. Let
$h_e(k)$
denote the number of length
$k$
paths in the replacement of
$e$
and let
$\alpha _k = \sum _{e \in E(H)} h_e(k)/\binom{h}{2}$
. Then we have
\begin{equation*}t_{H'}(W) \geq \int _{[0, 1]^h}\prod _{k \geq 1} \prod _{ij \in \binom{V(H)}{2}} W^k(x_i, x_j)^{\alpha _k} \prod _{i \in [h]} \mathrm{d}x_i.\end{equation*}
Proof. By integrating variables corresponding to
$V(H') \setminus V(H)$
first, we have
\begin{equation*} t_{H'}(W) = \int _{[0, 1]^h}\prod _{k \geq 1} \prod _{ij \in \binom{V(H)} {2}} W^k(x_{i}, x_{j})^{h_{ij}(k)} \prod _{i \in [h]} \mathrm{d}x_i,\end{equation*}
where we set
$h_{ij}(k)=0$
when
$ij$
is not an edge of
$H$
. We now observe that for any permutation
$\varphi\,:\,V(H) \rightarrow V(H)$
, there holds
\begin{equation*} t_{H'}(W) = \int _{[0, 1]^h}\prod _{k \geq 1} \prod _{ij \in \binom{V(H)}{2}} W^k(x_{\varphi (i)}, x_{\varphi (j)})^{h_{ij}(k)} \prod _{i \in [h]} \mathrm{d}x_i,\end{equation*}
as it is equivalent to the relabelling of vertices of
$H$
. By taking product over all permutations
$\varphi \,:\,[h] \rightarrow [h]$
and applying Hölder’s inequality(Theorem 2.8) with
$p_k=1$
for all
$k \in [h!]$
and
$r=1/h!$
, we have
\begin{align*} t_{H'}(W)^{h!} & = \prod _{\varphi \,:\,[h] \rightarrow [h]} \left (\int _{[0, 1]^h}\prod _{k \geq 1} \prod _{ij \in \binom{V(H)}{2}} W^k(x_{\varphi (i)}, x_{\varphi (j)})^{h_{ij}(k)} \prod _{i \in [h]} \mathrm{d}x_i \right ) \\ & \geq \left (\int _{[0, 1]^h} \prod _{\varphi \,:\,[h] \rightarrow [h]}\prod _{uv \in E(H)} \prod _{k \geq 1} W^k(x_{\varphi (u)}, x_{\varphi (v)})^{h_{uv}(k)/h!} \prod _{i \in [h]} \mathrm{d}x_i \right )^{h!}. \end{align*}
Now fix distinct
$i, j \in V(H)$
and
$k\in \mathbb{N}$
. Consider the power of
$W^k(x_i, x_j)$
in the product
\begin{equation*}\prod _{\varphi \,:\,[h] \rightarrow [h]}\prod _{uv \in E(H)} W^k(x_{\varphi (u)}, x_{\varphi (v)})^{h_{uv}(k)/h!}.\end{equation*}
For any distinct
$i', j' \in V(H)$
, the number of permutations
$\varphi$
that maps
$i'$
to
$i$
and
$j'$
to
$j$
is
$(h-2)!$
. As
$W^k(x_i, x_j) = W^k(x_j, x_i)$
, we have
$2(h-2)!$
permutations such that
$W^k(x_i, x_j) = W^k(x_{\varphi (i')},$
$ x_{\varphi (j')})$
. Therefore, the power of
$W^k(x_i, x_j)$
in
$\prod _{\varphi \,:\,[h] \rightarrow [h]}\prod _{uv \in E(H)}W^k(x_{\varphi (u)}, x_{\varphi (v)})^{h_{uv}(k)/h!}$
is
\begin{equation*}\sum _{u'v' \in \binom{V(H)}{2}} h_{u'v'}(k) 2(h-2)!/h! = \alpha _k.\end{equation*}
Hence we have
\begin{equation*}t_{H'}(W)^{h!} \geq \left (\int _{[0, 1]^h}\prod _{k \geq 1} \prod _{ij \in \binom{V(H)} {2}} W^k(x_i, x_j)^{\alpha _k} \prod _{i \in [h]} \mathrm{d}x_i\right )^{h!},\end{equation*}
which completes the proof.
We are now ready to prove Theorem 1.4.
Proof of Theorem 1.4. Let
$H, H'$
be as in the statement and
$W$
be a
$d$
-regular graphon. Let
$\alpha _k = \sum _{e \in E(H)} h_e(k)/{\binom{h}{2}}$
. Note that by the assumption,
$\alpha _{2k-1}=0$
for every
$k \geq 1$
. By Lemma 3.4, we have
\begin{align} t_{H'}(W) \geq \int _{[0, 1]^h}\prod _{k \geq 1} \prod _{ij \in \binom{V(H)} {2}} W^{2k}(x_i, x_j)^{\alpha _{2k}} \prod _{i \in [h]} \mathrm{d}x_i. \end{align}
For the first case, if all of
$\sum _{e \in E(H)} h_e(2k)$
is divisible by
$\binom{h}{2}$
, then
$\alpha _{2k}$
is an integer for all
$k \geq 1$
. Let
$\Theta$
be a theta graph which consists of
$\alpha _{2k}$
paths of length
$2k$
for all
$k\ge 1$
and let
$F$
be a graph obtained by replacing each edge of
$K_h$
by
$\Theta$
. Then by Theorem 1.3,
$F$
is Sidorenko. Noting that
$e(F)=\binom{h}{2}\sum _{k} \alpha _{2k}\cdot 2k=\sum _{k}\sum _{e \in E(H)} h_e(2k)=e(H')$
, we then have
\begin{equation*}t_{H'}(W) \geq \int _{[0, 1]^h}\prod _{k \geq 1} \prod _{ij \in \binom{V(H)} {2}} W^{2k}(x_i, x_j)^{\alpha _{2k}} \prod _{i \in [h]} \mathrm{d}x_i = t_F(W) \geq d^{e(F)} = d^{e(H')}.\end{equation*}
Hence
$H'$
is Sidorenko.
We now consider the second case. If
$\sum _{e \in E(H)} h_e(2k)=0$
for all but one
$k$
and
$\sum _{e \in E(H)} h_e(2k) \geq \binom {h}{2}$
for such exceptional
$k$
, then
$\alpha _{2k} \geq 1$
for such exceptional
$k$
and
$\alpha _{2k}=0$
for all other
$k$
. Then (2) becomes
\begin{equation*}t_{H'}(W) \geq \int _{[0, 1]^h} \left (\prod _{ij \in \binom{V(H)}{2}} W^{2k}(x_i, x_j)\right )^{\alpha _{2k}} \prod _{i \in [h]} \mathrm{d}x_i\end{equation*}
for some
$k \geq 1$
. By applying Jensen’s inequality with
$x \mapsto x^{\alpha _k}$
, we have
\begin{align*} \int _{[0, 1]^h} \left (\prod _{ij \in \binom{V(H)}{2}} W^{2k}(x_i, x_j)\right )^{\alpha _{2k}} \prod _{i \in [h]} \mathrm{d}x_i &\geq \left (\int _{[0, 1]^h} \prod _{ij \in \binom{V(H)}{2} } W^{2k}(x_i, x_j) \prod _{i \in [h]} \mathrm{d}x_i\right )^{\alpha _{2k}}\\ &= t_{K_h}(W^{2k})^{\alpha _{2k}} \geq d^{e(H')}, \end{align*}
where the last inequality comes from the fact that
$(2k-1)$
-subdivision of
$K_h$
is Sidorenko. Therefore,
$H'$
is Sidorenko.
3.3 Flowers are KNRS
In this section, we prove Theorem 1.8.
Proof of Theorem 1.8. Let
$W$
be a
$d$
-locally dense graphon and let
$H$
be a flower. Let
$C_1, \ldots , C_{r+s}$
be cycles of
$H$
and
$v$
be the vertex that all cycles intersect where
$C_1, \ldots , C_r$
has length
$2\ell _1+1, \ldots , 2\ell _r+1$
and
$C_{r+1}, \ldots , C_{r+s}$
has length
$2\ell _{r+1}, \ldots , 2\ell _{r+s}$
. Then we have
\begin{align*} t_H(W) &= \int _{[0, 1]}\left (\int _{[0, 1]^{2r}} \prod _{i \in [r]} W^{\ell _i}(x, y_i)W^{\ell _i}(x, y'_i) W(y_i, y'_i) \prod _{i \in [r]} \mathrm{d}y_i \mathrm{d}y'_i \right )\\ &\quad \cdot \left (\int _{[0, 1]^{s}} \prod _{r+1 \leq i \leq r+s} W^{\ell _i}(x, z_i)^2 \prod _{r+1 \leq i \leq r+s} \mathrm{d}z_i \right ) \mathrm{d}x. \end{align*}
By Lemma 2.6, we have
\begin{equation*}\int _{[0, 1]^{2}} W^{\ell _i}(x, y_i)W^{\ell _i}(x, y'_i) W(y_i, y'_i) \mathrm{d}y_i\mathrm{d}y'_i \geq d \left (\int _{[0, 1]} W^{\ell _i}(x, y) \mathrm{d}y \right )^2\end{equation*}
for every
$x \in [0, 1]$
and
$i \in [r]$
. Also, by Jensen’s inequality,
\begin{equation*}\int _{[0, 1]^{s}} \prod _{r+1 \leq i \leq r+s} W^{\ell _i}(x, z_i)^2 \prod _{r+1 \leq i \leq r+s} \mathrm{d}z_i \geq \prod _{r+1 \leq i \leq r+s} \left (\int _{[0, 1]} W^{\ell _i}(x, y) \mathrm{d}y \right )^2.\end{equation*}
Thus, using Jensen’s inequality again we have
\begin{align*} t_H(W) & \geq d^r \int _{[0, 1]} \prod _{i \in [r]} \left (\int _{[0, 1]} W^{\ell _i}(x, y) \mathrm{d}y \right )^2 \prod _{r+1 \leq i \leq r+s} \left (\int _{[0, 1]} W^{\ell _i}(x, y) \mathrm{d}y \right )^2 \mathrm{d}x \\ & \geq d^r \left (\int _{[0, 1]} \prod _{i \in [r+s]} \left (\int _{[0, 1]} W^{\ell _i}(x, y)\mathrm{d}y \right ) \mathrm{d}x \right )^2 \\ & = d^r \left (\int _{[0, 1]} \int _{[0, 1]^{r+s}} \prod _{i \in [r+s]}W^{\ell _i}(x, y_i) \prod _{i \in [r+s]}\mathrm{d}y_i \mathrm{d}x \right )^2. \end{align*}
Let
$F$
be a tree obtained by attaching
$r+s$
paths of length
$\ell _1, \ldots , \ell _{r+s}$
to a single vertex. Then we observe that
$t_F(W) = \int _{[0, 1]} \int _{[0, 1]^{r+s}} \prod _{i \in [r+s]}W^{\ell _i}(x, y_i) \prod _{i \in [r+s]}\mathrm{d}y_i \mathrm{d}x$
. Also,
$F$
is Sidorenko as it is a tree. Therefore,
\begin{equation*}t_H(W) \geq d^r t_F(W)^2 \geq d^r d^{2e(F)} = d^{r+\sum _{i \in [r+s]} 2\ell _i} = d^{e(H)},\end{equation*}
so
$H$
is KNRS as desired.
4. Concluding remarks
In this paper, we provide a large class of subdivisions that are Sidorenko. We believe that our results can be extended to a more general setting.
Conjecture 4.1.
Let
$H$
be a Sidorenko graph and
$H'$
is a subdivision of
$H$
. Then
$H'$
is Sidorenko whenever it is bipartite.
We remark that the uniform case of this conjecture can be proved easily.
Proposition 4.2.
Let
$H$
be a Sidorenko graph and
$H'$
be an
$\ell$
-subdivision of
$H$
. Then
$H'$
is Sidorenko.
Proof. Let
$W$
be a
$d$
-regular graphon. Then we obtain
\begin{align*} t_{H'}(W) = \int _{[0, 1]^{v(H)}} \prod _{ij \in E(H)} W^{\ell +1}(x_i, x_j) \prod _{i \in [v(H)]} \mathrm{d}x_i \geq t_{K_2}(W^{\ell +1})^{e(H)} \geq d^{e(H')}, \end{align*}
where the last inequality comes from the fact that the path of length
$\ell +1$
is Sidorenko.
Another special case of Conjecture 4.1 is to consider odd generalised theta graphs. We say a simple graph
$H$
is an odd generalised theta graph if it can be obtained by adding internally disjoint odd paths between two fixed root vertices
$u$
and
$v$
. The Sidorenko property of odd generalised theta graphs can be derived from the result by Li and Szegedy [Reference Li and Szegedy10] or from the strong tree-decomposition result by Conlon, Kim, Lee and Lee [Reference Conlon, Kim, Lee and Lee5].
Proposition 4.3. Every odd generalised theta graph is Sidorenko.
Proof Sketch. For
$k\geq 2$
and odd integers
$\ell _1\geq \ell _2\geq \cdots \geq \ell _k\geq 1$
, let
$T_0$
be a tree obtained by attaching
$k$
paths of lengths
$\ell _k, \frac {\ell _1-\ell _k}{2}, \frac {\ell _2-\ell _k}{2},\ldots ,\frac {\ell _{k-1}-\ell _k}{2}$
to a vertex
$s$
. Assume the other ends of the paths are
$t,x_1,x_2,\ldots ,x_{k-1}$
, respectively. Particularly, if
$\ell _i=\ell _k$
for some
$1\le i\le k-1$
, then
$x_i=s$
. Let
$T_i$
be a path of length
$\frac {\ell _i+\ell _k}{2}$
connecting
$t$
and
$x_i$
such that the internal vertices of
$T_i$
is disjoint with
$\cup _{j=0}^{i-1} V(T_j)$
. Let
$\Theta =\cup _{i=0}^{k-1} T_i$
. Obviously,
$\Theta$
is an odd generalised theta graph for which those paths connecting two roots are of lengths
$\ell _1,\ell _2,\ldots ,\ell _k$
, respectively. Let
$\mathcal{F}=\{T_0,T_1,\ldots ,T_{k-1}\}$
and let
$\mathcal{T}$
be a
$(k-1)$
-edge star on
$\mathcal{F}$
with
$T_0$
being the
$(k-1)$
-degree vertex. Then
$(\mathcal{F},\mathcal{T})$
is a tree decomposition of
$\Theta$
. Such a decomposition indicates that
$\Theta$
is strongly tree-decomposable and therefore is Sidorenko according to Theorem 1.2 in the paper of Conlon-Kim-Lee-Lee [Reference Conlon, Kim, Lee and Lee5].
Acknowledgements
We would like to thank Joonkyung Lee for fruitful discussions and thank Jon Noel for telling us about Proposition 4.3.
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