arXiv subjects

Joonkyung Lee

Publications and source records attributed to Joonkyung Lee.

33 records · Page 2Linked to original sources

arXiv2022

On graph norms for complex-valued functions

For any given graph $H$, one may define a natural corresponding functional $\|.\|_H$ for real-valued functions by using homomorphism density. One may also extend this to complex-valued functions, once $H$ is paired with a $2$-edge-colouring $α$ to assign conjugates. We say that $H$ is real-norming (resp. complex-norming) if $\|.\|_H$ (resp. $\|.\|_{H,α}$ for some $α$) 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 $H$ 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 $2$-edge-colouring $α$ 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.

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arXiv2021

Majority dynamics on sparse random graphs

Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $\pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnel, Tamuz and Tan conjectured that, in the Erdős--Rényi random graph $G(n,p)$, the random initial $\pm 1$-assignment converges to a $99\%$-agreement with high probability whenever $p=ω(1/n)$. This conjecture was first confirmed for $p\geqλn^{-1/2}$ for a large constant $λ$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< λn^{-1/2}$. We break this $Ω(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $λ' n^{-3/5}\log n \leq p \leq λn^{-1/2}$ with a large constant $λ'>0$.

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arXiv2021

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 $A \cup B$ 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 $A \cup B$, 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 $H$.

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arXiv2020

Ramsey games near the critical threshold

A well-known result of Rödl and Ruciński states that for any graph $H$ there exists a constant $C$ such that if $p \geq C n^{- 1/m_2(H)}$, then the random graph $G_{n,p}$ is a.a.s. $H$-Ramsey, that is, any $2$-colouring of its edges contains a monochromatic copy of $H$. Aside from a few simple exceptions, the corresponding $0$-statement also holds, that is, there exists $c>0$ such that whenever $p\leq cn^{-1/m_2(H)}$ the random graph $G_{n,p}$ is a.a.s. not $H$-Ramsey. We show that near this threshold, even when $G_{n,p}$ is not $H$-Ramsey, it is often extremely close to being $H$-Ramsey. More precisely, we prove that for any constant $c > 0$ and any strictly $2$-balanced graph $H$, if $p \geq c n^{-1/m_2(H)}$, then the random graph $G_{n,p}$ a.a.s. has the property that every $2$-edge-colouring without monochromatic copies of $H$ cannot be extended to an $H$-free colouring after $ω(1)$ extra random edges are added. This generalises a result by Friedgut, Kohayakawa, Rödl, Ruciński and Tetali, who in 2002 proved the same statement for triangles, and addresses a question raised by those authors. We also extend a result of theirs on the three-colour case and show that these theorems need not hold when $H$ is not strictly $2$-balanced.

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arXiv2020

More on the extremal number of subdivisions

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing $K'_{s,t}$ for the subdivision of the bipartite graph $K_{s,t}$, we show that $\mathrm{ex}(n, K'_{s,t}) = O(n^{3/2 - \frac{1}{2s}})$. This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for $t$ sufficiently large in terms of $s$. Second, for any integers $s, k \geq 1$, we show that $\mathrm{ex}(n, L) = Θ(n^{1 + \frac{s}{sk+1}})$ for a particular graph $L$ depending on $s$ and $k$, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number $r \in (1,2)$ is realisable in the sense that $\mathrm{ex}(n,H) = Θ(n^r)$ for some appropriate graph $H$, giving infinitely many new realisable exponents and implying that $1 + 1/k$ is a limit point of realisable exponents for all $k \geq 1$. Writing $H^k$ for the $k$-subdivision of a graph $H$, this result also implies that for any bipartite graph $H$ and any $k$, there exists $δ> 0$ such that $\mathrm{ex}(n,H^{k-1}) = O(n^{1 + 1/k - δ})$, partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph $H$ with maximum degree $r$ on one side which does not contain $C_4$ as a subgraph satisfies $\mathrm{ex}(n, H) = o(n^{2 - 1/r})$.

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arXiv2020

Convex graphon parameters and graph norms

Sidorenko's conjecture states that the number of copies of a bipartite graph $H$ in a graph $G$ is asymptotically minimised when $G$ is a quasirandom graph. A notorious example where this conjecture remains open is when $H=K_{5,5}\setminus C_{10}$. It was even unknown whether this graph possesses the strictly stronger, weakly norming property. We take a step towards understanding the graph $K_{5,5}\setminus C_{10}$ by proving that it is not weakly norming. More generally, we show that 'twisted' blow-ups of cycles, which include $K_{5,5}\setminus C_{10}$ and $C_6\square K_2$, are not weakly norming. This answers two questions of Hatami. The method relies on the analysis of Hessian matrices defined by graph homomorphisms, by using the equivalence between the (weakly) norming property and convexity of graph homomorphism densities. We also prove that $K_{t,t}$ minus a perfect matching, proven to be weakly norming by Lovász, is not norming for every $t>3$.

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arXiv2019

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 $L^p$, $p\geq e(H)$, 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 not 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 an error in the original statement of the aforementioned theorem by Hatami.

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arXiv2019

On some graph densities in locally dense graphs

The Kohayakawa-Nagle-Rödl-Schacht conjecture roughly states that every sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. Firstly, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Secondly, we prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies the conjecture. We also prove an analogous result with odd cycles replacing complete multipartite graphs.

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arXiv2019

Odd cycles in subgraphs of sparse pseudorandom graphs

We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let $Γ$ be an $(n,d,λ)$-graph, i.e., $n$-vertex, $d$-regular graphs with all nontrivial eigenvalues in the interval $[-λ,λ]$. Krivelevich, Lee, and Sudakov conjectured that, whenever $λ^{2k-1}\ll d^{2k}/n$, every subgraph $G$ of $Γ$ with $(1/2+o(1))e(Γ)$ edges contains an odd cycle $C_{2k+1}$. Aigner-Horev, Hàn, and the third author proved a weaker statement by allowing an extra polylogarithmic factor in the assumption $λ^{2k-1}\ll d^{2k}/n$, but we completely remove it and hence settle the conjecture. This also generalises Sudakov, Szabo, and Vu's Turán-type theorem for triangles. Secondly, we obtain a Ramsey multiplicity result for odd cycles. Namely, in the same range of parameters, we prove that every 2-edge-colouring of $Γ$ contains at least $(1-o(1))2^{-2k}d^{2k+1}$ monochromatic copies of $C_{2k+1}$. Both results are asymptotically best possible by Alon and Kahale's construction of $C_{2k+1}$-free pseudorandom graphs.

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arXiv2019

On the extremal number of subdivisions

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if $H$ is a bipartite graph with maximum degree $r$ on one side, then there is a constant $C$ such that every graph with $n$ vertices and $C n^{2 - 1/r}$ edges contains a copy of $H$. This result is tight up to the constant when $H$ contains a copy of $K_{r,s}$ with $s$ sufficiently large in terms of $r$. We conjecture that this is essentially the only situation in which Füredi's result can be tight and prove this conjecture for $r = 2$. More precisely, we show that if $H$ is a $C_4$-free bipartite graph with maximum degree $2$ on one side, then there are positive constants $C$ and $δ$ such that every graph with $n$ vertices and $C n^{3/2 - δ}$ edges contains a copy of $H$. This answers a question of Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.

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arXiv2018

Sidorenko's conjecture for higher tree decompositions

This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader class, comparable to that given in work of Szegedy, through further iteration.

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arXiv2018

Some advances on Sidorenko's conjecture

A bipartite graph $H$ is said to have Sidorenko's property if the probability that the uniform random mapping from $V(H)$ to the vertex set of any graph $G$ is a homomorphism is at least the product over all edges in $H$ of the probability that the edge is mapped to an edge of $G$. In this paper, we provide three distinct families of bipartite graphs that have Sidorenko's property. First, using branching random walks, we develop an embedding algorithm which allows us to prove that bipartite graphs admitting a certain type of tree decomposition have Sidorenko's property. Second, we use the concept of locally dense graphs to prove that subdivisions of certain graphs, including cliques, have Sidorenko's property. Third, we prove that if $H$ has Sidorenko's property, then the Cartesian product of $H$ with an even cycle also has Sidorenko's property.

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arXiv2017

Finite reflection groups and graph norms

Given a graph $H$ on vertex set $\{1,2,\cdots, n\}$ and a function $f:[0,1]^2 \rightarrow \mathbb{R}$, define \begin{align*} \|f\|_{H}:=\left\vert\int \prod_{ij\in E(H)}f(x_i,x_j)dμ^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*} where $μ$ is the Lebesgue measure on $[0,1]$. We say that $H$ is norming if $\|\cdot\|_H$ is a semi-norm. A similar notion $\|\cdot\|_{r(H)}$ is defined by $\|f\|_{r(H)}:=\||f|\|_{H}$ and $H$ is said to be weakly norming if $\|\cdot\|_{r(H)}$ 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 conjecture.

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arXiv2014

Two Approaches to Sidorenko's Conjecture

Sidorenko's conjecture states that for every bipartite graph $H$ on $\{1,\cdots,k\}$, $\int \prod_{(i,j)\in E(H)} h(x_i, y_j) dμ^{|V(H)|} \ge \left( \int h(x,y) \,dμ^2 \right)^{|E(H)|}$ holds, where $μ$ is the Lebesgue measure on $[0,1]$ and $h$ is a bounded, non-negative, symmetric, measurable function on $[0,1]^2$. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph $H$ to a graph $G$ is asymptotically at least the expected number of homomorphisms from $H$ to the Erdős-Rényi random graph with the same expected edge density as $G$. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph $H$ with bipartition $A \cup B$ is tree-arrangeable if neighborhoods of vertices in $A$ have a certain tree-like structure. We show that Sidorenko's conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's conjecture holds if there are two vertices $a_1, a_2$ in $A$ such that each vertex $a \in A$ satisfies $N(a) \subseteq N(a_1)$ or $N(a) \subseteq N(a_2)$, and also implies a recent result of Conlon, Fox, and Sudakov \cite{CoFoSu}. Second, if $T$ is a tree and $H$ is a bipartite graph satisfying Sidorenko's conjecture, then it is shown that the Cartesian product $T \Box H$ of $T$ and $H$ also satisfies Sidorenko's conjecture. This result implies that, for all $d \ge 2$, the $d$-dimensional grid with arbitrary side lengths satisfies Sidorenko's conjecture.

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arXiv2010

Rank-width of Random Graphs

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i) if 0 1, then rw(G(n,p)) > r n for some r = r(c), and (iv) if p <= c/n and c<1, then rw(G(n,p)) <=2. As a corollary, we deduce that G(n,p) has linear tree-width whenever p=c/n for each c>1, answering a question of Gao (2006).

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