arXiv subjects

Mihyun Kang

Publications and source records attributed to Mihyun Kang.

At least 19 recordsLinked to original sources

arXiv2026

Universal behaviour of majority bootstrap percolation on high-dimensional geometric graphs

Majority bootstrap percolation is a monotone cellular automaton that can be thought of as a model of infection spreading in networks. Starting with an initially infected set, new vertices become infected once at least half of their neighbours are infected. The average case behaviour of this process was studied on the $n$-dimensional hypercube by Balogh, Bollobás and Morris, who showed that there is a phase transition as the typical density of the initially infected set increases: For small enough densities the spread of infection is typically local, whereas for large enough densities typically the whole graph eventually becomes infected. Perhaps surprisingly, they showed that the critical window in which this phase transition occurs does not contain $p=1/2$, and they gave bounds on its width on a finer scale. In this paper we consider the majority bootstrap percolation process on a class of high-dimensional geometric graphs which includes many of the graph families on which percolation processes are typically considered, such as grids, tori and Hamming graphs, as well as other well-studied families of graphs such as (bipartite) Kneser graphs, including the odd graph and the middle layer graph. We show similar quantitative behaviour in terms of the location and width of the critical window for the majority bootstrap percolation process on this class of graphs.

math.CO↗

arXiv2026

Uniqueness and locality of the ground state of the disordered Monomer-Dimer models on independently weighted Unimodular Bienaymé-Galton-Watson trees

Consider a finite graph $G=(V(G),E(G))$ and two continuous weight distributions $ω$ and $ξ$, for which we only assume that $ξ$ is lower bounded. Next, independently draw weights $(w(e))_{e \in E(G)}$ with distribution $ω$ on edges and $(x(v))_{v \in V(G)}$ with distribution $ξ$ on vertices. The ground state of the monomer-dimer model on the weighted graph $G$ is a collection of edges (dimers) and vertices (monomers) such that every vertex is included in at most one monomer or dimer, and such that the sum of weights on its dimers and monomers is maximised. Take $(G_n,o_n)_{n \in \mathbb{N}}$ to be a sequence of random rooted weighted graphs that converges locally to an independently weighted unimodular Bienaymé-Galton-Watson tree $(\mathbb{T},o)$ with vertex-weight distribution $ξ$ and edge-weight distribution $ω$ . By proving that the ground state of the monomer-dimer model on the tree $(\mathbb{T},o)$ is almost surely unique and locally approximable, we prove that the ground state of the monomer-dimer model on $(G_n,o_n)$ must converge locally to the ground state of the monomer-dimer model on $(\mathbb{T},o)$. This also implies a strong decorrelation property on monomer-dimer models on unimodular Bienaymé-Galton-Watson trees.

math.PR↗

arXiv2026

A short proof of a central limit theorem for the order of the giant component and $k$-core

In this note we outline a new and simple approach to proving central limit theorems for various 'global' graph parameters which have robust 'local' approximations, using the Efron--Stein inequality, which relies on a combinatorial analysis of the stability of these approximations under resampling an edge. As an application, we give short proofs of a central limit theorem for the order of the giant component and of the $k$-core for sparse random graphs.

math.CO↗

arXiv2026

Counting independent sets in percolated graphs via the Ising model

Given a graph $G$, we form a random subgraph $G_p$ by including each edge of $G$ independently with probability $p$. We provide an asymptotic expansion of the expected number of independent sets in random subgraphs of regular bipartite graphs satisfying certain vertex-isoperimetric properties, extending the work of Kronenberg and Spinka on the percolated hypercube. Combining graph containers with the cluster expansion from statistical physics, we give an expansion of the partition function of the Ising model in certain range of the parameters. Among other applications, we obtain results for even tori of growing side-length. As a tool, we prove a refined container lemma for the Ising model, which mildly improves recent bounds of Jenssen, Malekshahian, and Park.

math.CO↗

arXiv2026

Counting subgraphs in bounded-size Achlioptas processes

Achlioptas processes such as the Bohman--Frieze process are much harder to analyse than the classical Erdős--Rényi process, due to the dependence between edges added at different stages. This dependence means that most analysis so far is dynamic, often based on the differential equation method. In the Erdős--Rényi case there is an alternative static approach, pioneered by Erdős, Rényi and Bollobás, based on evaluating the expectation (and higher moments) of various subgraph counts, and using this to study the component structure. Here we show that this latter approach can be applied (with some complications) to the Bohman--Frieze process. For example, we are able to show that the expected number $μ_{k,t,n}$ of $k$-vertex tree components after $tn$ steps satisfies (essentially) $μ_{k,t,n}=c_{k,t}n(1+O(k/\sqrt{n}))$. Our method gives a very complicated formula for $c_{k,t}$, which seems to be unusable. However, since $c_{k,t}$ does not depend on $n$, we may use recent results obtained by the differential equation method and branching process analysis to find the asymptotics of $c_{k,t}$ as $k\to\infty$. The latter results also give a formula for $μ_{k,t,n}$ of the form $c_{k,t}n$ plus an error term, with a much more usable description of $c_{k,t}$ but a much worse error term. We combine the best of both worlds to prove a number of new results about the process near criticality. In particular, we obtain extremely sharp bounds on the size of the largest non-giant component near criticality, including the limiting distribution of its fluctuations.

math.PR↗

arXiv2026

Block-weighted random graphs: planar and beyond

We investigate random connected graphs from a block-stable class whose distribution is weighted based on the number of $2$-connected components, or blocks. This includes the class of planar graphs. For this, we develop a notion of a decorated block tree. Following similar ideas to Fleurat and the second author on block-weighted planar maps, we find a phase transition in the singular behaviour of the appropriate generating function and in the typical structure of the block tree. Moreover, for certain block-stable classes (including planar graphs), we obtain precise enumeration results and determine also the typical sizes of the largest blocks in subcritical, critical, and supercritical regimes. It strengthens previously known results on block sizes in uniform random planar graphs.

math.CO↗

arXiv2026

Sampling from the antiferromagnetic Ising model on bipartite, regular expander graphs

The antiferromagnetic Ising model samples subsets of vertices of a graph with weight decaying exponentially in the number of edges induced. We study the problem of sampling from this model on the class of bipartite, regular graphs with good vertex expansion. We show that a natural sampler, namely the Glauber dynamics, mixes exponentially slowly in a wide range of parameters. On the other hand, we give an efficient alternative algorithm for sampling from the Ising model and an FPTAS for its partition function, using polymer models and the cluster expansion method.

math.CO↗

arXiv2025

Fragile minor-monotone parameters under random edge perturbation

We conduct a quantitative analysis of how many random edges need to be added to a base graph $H$ in order to significantly increase natural minor-monotone graph parameters of the resulting graph $R$. Specifically, we show that if $R$ is obtained from a connected graph $H$ by adding only a few random edges, the tree-width, genus, and Hadwiger number of $R$ become very large, irrespective of the structure of $H$.

math.CO↗

arXiv2025

Optimally building spanning graphs in semirandom graph processes

The semirandom graph process constructs a graph $G$ in a series of rounds, starting with the empty graph on $n$ vertices. In each round, a player is offered a vertex $v$ chosen uniformly at random, and chooses an edge on $v$ to add to $G$. The player's aim is to make $G$ satisfy some property as quickly as possible. Our interest is in the property that $G$ contain a given $n$-vertex graph $H$ with maximum degree $Δ$. In 2021, Ben-Eliezer, Gishboliner, Hefetz and Krivelevich showed that there is a semirandom strategy that achieves this, with probability tending to 1 as $n$ tends to infinity, in $(1 + o_Δ(1)) \frac{3 Δn}{2}$ rounds, where $o_Δ(1)$ is a function that tends to $0$ as $Δ$ tends to infinity. We improve this to $(1 + o_Δ(1)) \frac{Δn}{2}$, which can be seen to be asymptotically optimal in $Δ$. We show the same result for a variant of the semirandom graph process, namely the semirandom tree process introduced by Burova and Lichev, where in each round the player is offered the edge set of a uniformly chosen tree on $n$ vertices, and chooses one edge to keep.

math.CO↗

arXiv2025

Belief Propagation Guided Decimation on Random k-XORSAT

We analyse the performance of Belief Propagation Guided Decimation, a physics-inspired message passing algorithm, on the random $k$-XORSAT problem. Specifically, we derive an explicit threshold up to which the algorithm succeeds with a strictly positive probability $Ω(1)$ that we compute explicitly, but beyond which the algorithm with high probability fails to find a satisfying assignment. In addition, we analyse a thought experiment called the decimation process for which we identify a (non-) reconstruction and a condensation phase transition. The main results of the present work confirm physics predictions from [RTS: J. Stat. Mech. 2009] that link the phase transitions of the decimation process with the performance of the algorithm, and improve over partial results from a recent article [Yung: Proc. ICALP 2024].

math.CO↗

arXiv2025

Warning Propagation on random graphs

Warning Propagation is a combinatorial message passing algorithm that unifies and generalises a wide variety of recursive combinatorial procedures. Special cases include the Unit Clause Propagation and Pure Literal algorithms for satisfiability as well as the peeling process for identifying the $k$-core of a random graph. Here we analyse Warning Propagation in full generality on the binomial random graph. We prove that under a mild stability assumption Warning Propagation converges rapidly. In effect, the analysis of the fixed point of the message passing process on a random graph reduces to analysing the process on a Galton-Watson tree. This result corroborates and generalises a heuristic first put forward by Pittel, Spencer and Wormald in their seminal $k$-core paper (JCTB 1996).

math.CO↗

arXiv2025

Majority bootstrap percolation on the permutahedron and other high-dimensional graphs

Majority bootstrap percolation is a model of infection spreading in networks. Starting with a set of initially infected vertices, new vertices become infected once half of their neighbours are infected. Balogh, Bollobás and Morris studied this process on the hypercube and showed that there is a phase transition as the density of the initially infected set increases. Generalising their results to a broad class of high-dimensional graphs, the authors of this work established similar bounds on the critical window, establishing a universal behaviour for these graphs. These methods necessitated an exponential bound on the order of the graphs in terms of their degrees. In this paper, we consider a slightly more restrictive class of high-dimensional graphs, which nevertheless covers most examples considered previously. Under these stronger assumptions, we are able to show that this universal behaviour holds in graphs of superexponential order. As a concrete and motivating example, we apply this result to the permutahedron, a symmetric high-dimensional graph of superexponential order which arises naturally in many areas of mathematics. Our methods also allow us to slightly improve the bounds on the critical window given in previous work, in particular in the case of the hypercube. Finally, the upper and lower bounds on the critical window depend on the maximum and minimum degree of the graph, respectively, leading to much worse bounds for irregular graphs. We also analyse an explicit example of a high-dimensional irregular graph, namely the Cartesian product of stars and determine the first two terms in the expansion of the critical probability, which in this case is determined by the minimum degree.

math.CO↗

arXiv2025

Cycle lengths in the percolated hypercube

Let $Q^d_p$ be the random subgraph of the $d$-dimensional binary hypercube obtained after edge-percolation with probability $p$. It was shown recently by the authors that, for every $\varepsilon > 0$, there is some $c = c(\varepsilon)>0$ such that, if $pd\ge c$, then typically $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. We strengthen this result to show that, under the same assumptions, typically $Q^d_p$ contains cycles of all even lengths between $4$ and $(1-\varepsilon)2^d$.

math.CO↗

arXiv2025

Nearly spanning cycle in the percolated hypercube

Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant $C=C(\varepsilon)>0$ such that, if $p\ge C/d$, then with high probability $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. This confirms a long-standing folklore conjecture, stated in particular by Condon, Espuny Díaz, Girão, Kühn, and Osthus [Hamiltonicity of random subgraphs of the hypercube, Mem. Amer. Math. Soc. 305 (2024), No. 1534].

math.CO↗

arXiv2025

Counting independent sets in expanding bipartite regular graphs

In this paper we provide an asymptotic expansion for the number of independent sets in a general class of regular, bipartite graphs satisfying some vertex-expansion properties, extending results of Jenssen and Perkins on the hypercube and strengthening results of Jenssen, Perkins and Potukuchi. More precisely, we give an expansion of the independence polynomial of such graphs using a polymer model and the cluster expansion. In addition to the number of independent sets, our results yields information on the typical structure of (weighted) independent sets in such graphs. The class of graphs we consider covers well-studied cases like the hypercube or the middle layers graph, and we show further that it includes any Cartesian product of bipartite, regular base graphs of bounded size. To this end, we prove strong bounds on the vertex expansion of bipartite and regular Cartesian product graphs, which might be of independent interest.

math.CO↗

arXiv2025

The law of the circumference of sparse binomial random graphs

There has been much interest in the distribution of the circumference, the length of the longest cycle, of a random graph $G(n,p)$ in the sparse regime, when $p = Θ\left(\frac{1}{n}\right)$. Recently, the first author and Frieze established a scaling limit for the circumference in this regime, along the way establishing an alternative 'structural' approximation for this parameter. In this paper, we give a central limit theorem for the circumference in this regime using a novel argument based on the Efron-Stein inequality, which relies on a combinatorial analysis of the effect of resampling edges on this approximation.

math.CO↗

arXiv2025

Universality of the matching number in percolated regular graphs

Fix a sequence of $d$-regular graphs $(G_d)_{d\in \mathbb{N}}$ and denote by $G_{d,p}$ the graph obtained from $G_d$ after edge-percolation with probability $p=c/d$, for a constant $c>0$. We prove a quantitative local convergence of $(G_{d,p})_{d\in \mathbb{N}}$. In combination with results of Bordenave, Lelarge and Salez, it implies that the rescaled matching number of $G_{d,p}$ is asymptotically equivalent to that of the binomial random graph $G(n,c/n)$.

math.CO↗

arXiv2024

Partitioning problems via random processes

There are a number of well-known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed graph has a 3-colouring such that for every vertex $v$, at most half of the out-neighbours of $v$ have the same colour as $v$. As another example, the internal partition conjecture, due to DeVos and to Ban and Linial, states that for every $d$, all but finitely many $d$-regular graphs have a partition into two nonempty parts such that for every vertex $v$, at least half of the neighbours of $v$ lie in the same part as $v$. We prove several results in this spirit: in particular, two of our results are that the majority colouring conjecture holds for Erdős-Rényi random directed graphs (of any density), and that the internal partition conjecture holds if we permit a tiny number of "exceptional vertices". Our proofs involve a variety of techniques, including several different methods to analyse random recolouring processes. One highlight is a "personality-changing" scheme: we "forget" certain information based on the state of a Markov chain, giving us more independence to work with.

math.CO↗