16 research outputs found
The step Sidorenko property and non-norming edge-transitive graphs
Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko
property, i.e., a quasirandom graph minimizes the density of H among all graphs
with the same edge density. We study a stronger property, which requires that a
quasirandom multipartite graph minimizes the density of H among all graphs with
the same edge densities between its parts; this property is called the step
Sidorenko property. We show that many bipartite graphs fail to have the step
Sidorenko property and use our results to show the existence of a bipartite
edge-transitive graph that is not weakly norming; this answers a question of
Hatami [Israel J. Math. 175 (2010), 125-150].Comment: Minor correction on page
The step Sidorenko property and non-norming edge-transitive graphs
Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a quasirandom multipartite graph minimizes the density of H among all graphs with the same edge densities between its parts; this property is called the step Sidorenko property. We show that many bipartite graphs fail to have the step Sidorenko property and use our results to show the existence of a bipartite edge-transitive graph that is not weakly norming; this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150]
Cut distance identifying graphon parameters over weak* limits
The theory of graphons comes with the so-called cut norm and the derived cut
distance. The cut norm is finer than the weak* topology. Dole\v{z}al and
Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin.
Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a
cut distance accumulation graphon can be pinpointed in the set of weak*
accumulation points as a minimizer of the entropy. Motivated by this, we study
graphon parameters with the property that their minimizers or maximizers
identify cut distance accumulation points over the set of weak* accumulation
points. We call such parameters cut distance identifying. Of particular
importance are cut distance identifying parameters coming from subgraph
densities, t(H,*). This concept is closely related to the emerging field of
graph norms, and the notions of the step Sidorenko property and the step
forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step
Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory
Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if
and only if it is step Sidorenko, and that if a graph is norming then it is
step forcing. Further, we study convexity properties of cut distance
identifying graphon parameters, and find a way to identify cut distance limits
using spectra of graphons. We also show that continuous cut distance
identifying graphon parameters have the "pumping property", and thus can be
used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming
graphs, and a new section 3.2 on index pumping in the regularity lemm
Finite reflection groups and graph norms
Given a graph on vertex set and a function , define \begin{align*} \|f\|_{H}:=\left\vert\int
\prod_{ij\in E(H)}f(x_i,x_j)d\mu^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*}
where is the Lebesgue measure on . We say that is norming if
is a semi-norm. A similar notion is defined by
and is said to be weakly norming if
is a norm. Classical results show that weakly norming graphs
are necessarily bipartite. In the other direction, Hatami showed that even
cycles, complete bipartite graphs, and hypercubes are all weakly norming. We
demonstrate that any graph whose edges percolate in an appropriate way under
the action of a certain natural family of automorphisms is weakly norming. This
result includes all previously known examples of weakly norming graphs, but
also allows us to identify a much broader class arising from finite reflection
groups. We include several applications of our results. In particular, we
define and compare a number of generalisations of Gowers' octahedral norms and
we prove some new instances of Sidorenko's conjecture.Comment: 29 page
Left-cut-percolation and induced-Sidorenko bigraphs
A Sidorenko bigraph is one whose density in a bigraphon is minimized
precisely when is constant. Several techniques of the literature to prove
the Sidorenko property consist of decomposing (typically in a tree
decomposition) the bigraph into smaller building blocks with stronger
properties. One prominent such technique is that of -decompositions of
Conlon--Lee, which uses weakly H\"{o}lder (or weakly norming) bigraphs as
building blocks. In turn, to obtain weakly H\"{o}lder bigraphs, it is typical
to use the chain of implications reflection bigraph cut-percolating
bigraph weakly H\"{o}lder bigraph. In an earlier result by the
author with Razborov, we provided a generalization of -decompositions,
called reflective tree decompositions, that uses much weaker building blocks,
called induced-Sidorenko bigraphs, to also obtain Sidorenko bigraphs.
In this paper, we show that "left-sided" versions of the concepts of
reflection bigraph and cut-percolating bigraph yield a similar chain of
implications: left-reflection bigraph left-cut-percolating bigraph
induced-Sidorenko bigraph. We also show that under mild hypotheses,
the "left-sided" analogue of the weakly H\"{o}lder property (which is also
obtained via a similar chain of implications) can be used to improve bounds on
another result of Conlon--Lee that roughly says that bigraphs with enough
vertices on the right side of each realized degree have the Sidorenko property.Comment: 42 pages, 5 figure
Two Remarks on Graph Norms
For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in Lp, p≥e(H), denoted by t(H, W). One may then define corresponding functionals ∥W∥H:=|t(H,W)|1/e(H) and ∥W∥r(H):=t(H,|W|)1/e(H), and say that H is (semi-)norming if ∥⋅∥H is a (semi-)norm and that H is weakly norming if ∥⋅∥r(H) is a norm. We obtain two results that contribute to the theory of (weakly) norming graphs. Firstly, answering a question of Hatami, who estimated the modulus of convexity and smoothness of ∥⋅∥H, we prove that ∥⋅∥r(H) is neither uniformly convex nor uniformly smooth, provided that H is weakly norming. Secondly, we prove that every graph H without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of H when studying graph norms. In particular, we correct a negligence in the original statement of the aforementioned theorem by Hatami
On graph norms for complex-valued functions
For any given graph , one may define a natural corresponding functional
for real-valued functions by using homomorphism density. One may also
extend this to complex-valued functions, once is paired with a
-edge-colouring to assign conjugates. We say that is
real-norming (resp. complex-norming) if (resp. for
some ) is a norm on the vector space of real-valued (resp.
complex-valued) functions. These generalise the Gowers octahedral norms, a
widely used tool in extremal combinatorics to quantify quasirandomness. We
unify these two seemingly different notions of graph norms in real- and
complex-valued settings. Namely, we prove that is complex-norming if and
only if it is real-norming and simply call the property norming. Our proof does
not explicitly construct a suitable -edge-colouring but obtains its
existence and uniqueness, which may be of independent interest. As an
application, we give various example graphs that are not norming. In
particular, we show that hypercubes are not norming, which resolves the last
outstanding problem posed in Hatami's pioneering work on graph norms.Comment: 33 page
Finite reflection groups and graph norms
Given a graph H on vertex set {1, 2, · · · , n} and a function f : [0, 1]2 → R, define
kfkH :=
Z Y
ij∈E(H)
f(xi
, xj )dµ|V (H)|
1/|E(H)|
,
where µ is the Lebesgue measure on [0, 1]. We say that H is norming if k·kH is a semi-norm.
A similar notion k·kr(H)
is defined by kfkr(H)
:= k|f|kH and H is said to be weakly norming if
k·kr(H)
is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In
the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes
are all weakly norming. We demonstrate that any graph whose edges percolate in an appropriate
way under the action of a certain natural family of automorphisms is weakly norming. This result
includes all previously known examples of weakly norming graphs, but also allows us to identify
a much broader class arising from finite reflection groups. We include several applications of our
results. In particular, we define and compare a number of generalisations of Gowers’ octahedral
norms and we prove some new instances of Sidorenko’s conjectur
Theory of combinatorial limits and extremal combinatorics
In the past years, techniques from different areas of mathematics have been successfully applied in extremal combinatorics problems. Examples include applications of number theory, geometry and group theory in Ramsey theory and analytical methods to different problems in extremal combinatorics.
By providing an analytic point of view of many discrete problems, the theory of combinatorial limits led to substantial results in many areas of mathematics and computer science, in particular in extremal combinatorics. In this thesis, we explore the connection between combinatorial limits and extremal combinatorics.
In particular, we prove that extremal graph theory problemsmay have unique optimal solutions with arbitrarily complex structure, study a property closely related to Sidorenko's conjecture, one of the most important open problems in extremal combinatorics, and prove a 30-year old conjecture of Gyori and Tuza regarding decomposing the edges of a graph into triangles and edges
Sidorenko's conjecture for blow-ups
A celebrated conjecture of Sidorenko and Erdős-Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion.
Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A ∪ B where the number of vertices in B of degree k satisfies a certain divisibility condition for each k. As a corollary, we have that for every bipartite graph H with bipartition A ∪ B, there is a positive integer p such that the blow-up H_(A)^(p) formed by taking p vertex-disjoint copies of H and gluing all copies of A along corresponding vertices satisfies the conjecture.
Another way of viewing this latter result is that for every bipartite H there is a positive integer p such that an L^(p)-version of Sidorenko's conjecture holds for H