arXiv subjects

Mihyun Kang

Publications and source records attributed to Mihyun Kang.

At least 37 records · Page 2Linked to original sources

arXiv2024

Large matchings and nearly spanning, nearly regular subgraphs of random subgraphs

Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $ε>0$, there exists a constant $C>0$ such that the following holds. Let $d\ge C$ be an integer, let $G$ be a $d$-regular graph and let $p\ge \frac{C}{d}$. Then, with probability tending to one as $|V(G)|$ tends to infinity, there exists a matching in $G_p$ covering at least $(1-ε)|V(G)|$ vertices. We further show that for a wide family of $d$-regular graphs $G$, which includes the $d$-dimensional hypercube, for any $p\ge \frac{\log^5d}{d}$ with probability tending to one as $d$ tends to infinity, $G_p$ contains an induced subgraph on at least $(1-o(1))|V(G)|$ vertices, whose degrees are tightly concentrated around the expected average degree $dp$.

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arXiv2024

Bootstrap percolation on the high-dimensional Hamming graph

In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get infected in subsequent rounds if they have at least $r$ infected neighbours. A graph $G$ \emph{percolates} if every vertex becomes eventually infected. A central problem in this process is to determine the critical probability $p_c(G,r)$, at which the probability that $G$ percolates passes through one half. In this paper, we study random $2$-neighbour bootstrap percolation on the $n$-dimensional Hamming graph $\square_{i=1}^n K_k$, which is the graph obtained by taking the Cartesian product of $n$ copies of the complete graph $K_k$ on $k$ vertices. We extend a result of Balogh and Bollobás [Bootstrap percolation on the hypercube, Probab. Theory Related Fields. 134 (2006), no. 4, 624-648. MR2214907] about the asymptotic value of the critical probability $p_c(Q^n,2)$ for random $2$-neighbour bootstrap percolation on the $n$-dimensional hypercube $Q^n=\square_{i=1}^n K_2$ to the $n$-dimensional Hamming graph $\square_{i=1}^n K_k$, determining the asymptotic value of $p_c\left(\square_{i=1}^n K_k,2\right)$, up to multiplicative constants (when $n \rightarrow \infty$), for arbitrary $k \in \mathbb N$ satisfying $2 \leq k\leq 2^{\sqrt{n}}$.

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arXiv2024

Cliques, Chromatic Number, and Independent Sets in the Semi-random Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. In this paper, we investigate the following three properties: containing a complete graph of order $k$, having the chromatic number at least $k$, and not having an independent set of size at least $k$.

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arXiv2024

Catching a robber on a random $k$-uniform hypergraph

The game of \emph{Cops and Robber} is usually played on a graph, where a group of cops attempt to catch a robber moving along the edges of the graph. The \emph{cop number} of a graph is the minimum number of cops required to win the game. An important conjecture in this area, due to Meyniel, states that the cop number of an $n$-vertex connected graph is $O(\sqrt{n})$. In 2016, Prałat and Wormald [Meyniel's conjecture holds for random graphs, Random Structures Algorithms. 48 (2016), no. 2, 396-421. MR3449604] showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreoever, Łuczak and Prałat [Chasing robbers on random graphs: Zigzag theorem, Random Structures Algorithms. 37 (2010), no. 4, 516-524. MR2760362] showed that on a $\log$-scale the cop number demonstrates a surprising \emph{zigzag} behaviour in dense regimes of the binomial random graph $G(n,p)$. In this paper, we consider the game of Cops and Robber on a hypergraph, where the players move along hyperedges instead of edges. We show that with high probability the cop number of the $k$-uniform binomial random hypergraph $G^k(n,p)$ is $O\left(\sqrt{\frac{n}{k}}\, \log n \right)$ for a broad range of parameters $p$ and $k$ and that on a $\log$-scale our upper bound on the cop number arises as the minimum of \emph{two} complementary zigzag curves, as opposed to the case of $G(n,p)$. Furthermore, we conjecture that the cop number of a connected $k$-uniform hypergraph on $n$ vertices is $O\left(\sqrt{\frac{n}{k}}\,\right)$.

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arXiv2024

Percolation on High-dimensional Product Graphs

We consider percolation on high-dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices. In the supercritical regime, our main result recovers the sharp asymptotic of the order of the largest component, and shows that all the other components are typically of order logarithmic in the number of vertices. In particular, we show that this phase transition is quantitatively similar to the one of the binomial random graph. This generalises the results of Ajtai, Komlós, and Szemerédi and of Bollobás, Kohayakawa, and Łuczak who showed that the $d$-dimensional hypercube, which is the $d$-fold Cartesian product of an edge, undergoes a phase transition quantitatively similar to the one of the binomial random graph.

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arXiv2024

Bootstrap Percolation on the Binomial Random $k$-uniform Hypergraph

We investigate the behaviour of $r$-neighbourhood bootstrap percolation on the binomial $k$-uniform random hypergraph $H_k(n,p)$ for given integers $k\geq 2$ and $r\geq 2$. In $r$-neighbourhood bootstrap percolation, infection spreads through the hypergraph, starting from a set of initially infected vertices, and in each subsequent step of the process every vertex with at least $r$ infected neighbours becomes infected. For our analysis the set of initially infected vertices is chosen uniformly at random from all sets of given size. In the regime $n^{-1}\ll n^{k-2}p \ll n^{-1/r}$ we establish a threshold such that if the number of initially infected vertices remains below the threshold, then with high probability only a few additional vertices become infected, while if the number of initially infected vertices exceeds the threshold then with high probability almost every vertex becomes infected. In fact we show that the probability of failure decreases exponentially.

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arXiv2024

A note on the width of sparse random graphs

In this note, we consider the width of a supercritical random graph according to some commonly studied width measures. We give short, direct proofs of results of Lee, Lee and Oum, and of Perarnau and Serra, on the rank- and tree-width of the random graph $G(n,p)$ when $p= \frac{1+ε}{n}$ for $ε> 0$ constant. Our proofs avoid the use, as a black box, of a result of Benjamini, Kozma and Wormald on the expansion properties of the giant component in this regime, and so as a further benefit we obtain explicit bounds on the dependence of these results on $ε$. Finally, we also consider the width of the random graph in the weakly supercritical regime, where $ε= o(1)$ and $ε^3n \to \infty$. In this regime, we determine, up to a constant multiplicative factor, the rank- and tree-width of $G(n,p)$ as a function of $n$ and $ε$.

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arXiv2024

Percolation on Irregular High-dimensional Product Graphs

We consider bond percolation on high-dimensional product graphs $G=\square_{i=1}^tG^{(i)}$, where $\square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $\frac{1}{d}$, where $d$ is the average degree of the host graph. In the supercritical regime, we strengthen Lichev's result by showing that the giant component is in fact unique, with all other components of order $o(|G|)$, and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev: firstly, we provide a construction showing that the requirement of bounded-degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdős-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper.

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arXiv2024

Isoperimetric Inequalities and Supercritical Percolation on High-dimensional Graphs

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component $L_1$ in supercritical percolation on these product graphs, that is, when $p=\frac{1+ε}{d}$, where $d$ is the degree of the product graph and $ε>0$ is a small enough constant. We show that typically $L_1$ has edge-expansion $Ω\left(\frac{1}{d\ln d}\right)$. Furthermore, we show that $L_1$ likely contains a linear-sized subgraph with vertex-expansion $Ω\left(\frac{1}{d\ln d}\right)$. These results are best possible up to the logarithmic factor in $d$. Using these likely expansion properties, we determine, up to small polylogarithmic factors in $d$, the likely diameter of $L_1$ as well as the typical mixing time of a lazy random walk on $L_1$. Furthermore, we show the likely existence of a path of length $Ω\left(\frac{n}{d\ln d}\right)$. These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of $L_1$ were previously only known to be polynomial in $d$.

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arXiv2024

Percolation through Isoperimetry

We provide a sufficient condition on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles the emergence of a giant component in the binomial random graph model $G(n,p)$. We further show that this condition is tight. More precisely, let $d=ω(1)$, let $ε>0$ be a small enough constant, and let $p \cdot d=1+ε$. We show that if $C$ is sufficiently large and $G$ is a $d$-regular $n$-vertex graph where every subset $S\subseteq V(G)$ of order at most $\frac{n}{2}$ has edge-boundary of size at least $C|S|$, then $G_p$ typically has a unique linear sized component, whose order is asymptotically $y(ε)n$, where $y(ε)$ is the survival probability of a Galton-Watson tree with offspring distribution Po$(1+ε)$. We further give examples to show that this result is tight both in terms of its dependence on $C$, and with respect to the order of the second-largest component. We also consider a more general setting, where we only control the expansion of sets up to size $k$. In this case, we show that if $G$ is such that every subset $S\subseteq V(G)$ of order at most $k$ has edge-boundary of size at least $d|S|$ and $p$ is such that $p\cdot d \geq 1 + ε$, then $G_p$ typically contains a component of order $Ω(k)$.

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arXiv2023

The sparse parity matrix

Let $\mathbf{A}$ be an $n\times n$-matrix over $\mathbb{F}_2$ whose every entry equals $1$ with probability $d/n$ independently for a fixed $d>0$. Draw a vector $\mathbf{y}$ randomly from the column space of $\mathbf{A}$. It is a simple observation that the entries of a random solution $\mathbf{x}$ to $\mathbf{A} x=\mathbf{y}$ are asymptotically pairwise independent, i.e., $\sum_{i \mathrm{e}$ the overlap concentrates on a single value once we condition on the matrix $\mathbf{A}$, while over the probability space of $\mathbf{A}$ its conditional expectation vacillates between two different values $α_*(d)<α^*(d)$, either of which occurs with probability $1/2+o(1)$. This bifurcated non-concentration result provides an instructive contribution to both the theory of random constraint satisfaction problems and of inference problems on random structures.

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arXiv2023

Component behaviour and excess of random bipartite graphs near the critical point

The binomial random bipartite graph $G(n,n,p)$ is the random graph formed by taking two partition classes of size $n$ and including each edge between them independently with probability $p$. It is known that this model exhibits a similar phase transition as that of the binomial random graph $G(n,p)$ as $p$ passes the critical point of $\frac{1}{n}$. We study the component structure of this model near to the critical point. We show that, as with $G(n,p)$, for an appropriate range of $p$ there is a unique `giant' component and we determine asymptotically its order and excess. We also give more precise results for the distribution of the number of components of a fixed order in this range of $p$. These results rely on new bounds for the number of bipartite graphs with a fixed number of vertices and edges, which we also derive.

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arXiv2023

On the chromatic number of graphons

We extend Bollobas' classical result on the chromatic number of a binomial random graph to the exchangeable random graph model $\mathcal{G}(n,W)$ defined by a graphon $W:[0,1]^2 \rightarrow [0,1]$, which is a symmetric measurable function. In the case when $W$ can be approximated by block graphons in $\mathcal{L}^{\infty}$-norm, we show that asymptotically optimal value of the number of colours required for $\mathcal{G}(n,W)$ is determined by colouring strategies that use a finite number of different types of colour classes. Furthermore, if $W$ is a block graphon with $k\times k$ blocks then $k$ types of colour classes are sufficient. We also show that if $W$ is block-increasing or block-Lipschitz then such colouring strategies that use $k$ types determine the chromatic number up to a multiplicative error of order $O(k^{-1})$.

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arXiv2023

On the chromatic number in the stochastic block model

We prove a generalisation of Bollobás' classical result on the asymptotics of the chromatic number of the binomial random graph to the stochastic block model. In addition, by allowing the number of blocks to grow, we determine the chromatic number in the Chung-Lu model. Our approach is based on the estimates for the weighted independence number, where weights are specifically designed to encapsulate inhomogeneities of the random graph.

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arXiv2023

The $k$-XORSAT threshold revisited

We provide a simplified proof of the random $k$-XORSAT satisfiability threshold theorem. As an extension we also determine the full rank threshold for sparse random matrices over finite fields with precisely $k$ non-zero entries per row. This complements a result from [Ayre, Coja-Oghlan, Gao, Müller: Combinatorica 2020]. The proof combines physics-inspired message passing arguments with a surgical moment computation.

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arXiv2022

On a question of Vera T. Sós about size forcing of graphons

The $k$-sample $\mathbb{G}(k,W)$ from a graphon $W:[0,1]^2\to [0,1]$ is the random graph on $\{1,\dots,k\}$, where we sample $x_1,\dots,x_k\in [0,1]$ uniformly at random and make each pair $\{i,j\}\subseteq \{1,\dots,k\}$ an edge with probability $W(x_i,x_j)$, with all these choices being mutually independent. Let the random variable $X_k(W)$ be the number of edges in $\mathbb{G}(k,W)$. Vera T. Sós asked in 2012 whether two graphons $U,W$ are necessarily weakly isomorphic if the random variables $X_k(U)$ and $X_k(W)$ have the same distribution for every integer $k\ge 2$. This question when one of the graphons $W$ is a constant function was answered positively by Endre Csóka and independently by Jacob Fox, Tomasz Łuczak and Vera T. Sós. Here we investigate the question when $W$ is a 2-step graphon and prove that the answer is positive for a 3-dimensional family of such graphons. We also present some related results.

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arXiv2022

The early evolution of the random graph process in planar graphs and related classes

We study the random planar graph process introduced by Gerke, Schlatter, Steger, and Taraz [The random planar graph process, Random Structures Algorithms 32 (2008), no. 2, 236--261; MR2387559]: Begin with an empty graph on $n$ vertices, consider the edges of the complete graph $K_n$ one by one in a random ordering, and at each step add an edge to a current graph only if the graph remains planar. They studied the number of edges added up to step $t$ for 'large' $t=ω(n)$. In this paper we extend their results by determining the asymptotic number of edges added up to step $t$ in the early evolution of the process when $t=O(n)$. We also show that this result holds for a much more general class of graphs, including outerplanar graphs, planar graphs, and graphs on surfaces.

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arXiv2022

Concentration of maximum degree in random planar graphs

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $[n]:=\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $m/n\leq 1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values. In contrast, this is not true anymore in the dense regime, when $m/n>1$, where the maximum degree of $P(n,m)$ is not concentrated on any subset of $[n]$ with bounded size.

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