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A polynomial-time approximation scheme for the maximal overlap of two independent Erdős–Rényi graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2024-03-07 Jian Ding, Hang Du, Shuyang Gong
For two independent Erdős–Rényi graphs G(n,p)$$ \mathbf{G}\left(n,p\right) $$, we study the maximal overlap (i.e., the number of common edges) of these two graphs over all possible vertex correspondence. We present a polynomial-time algorithm which finds a vertex correspondence whose overlap approximates the maximal overlap up to a multiplicative factor that is arbitrarily close to 1. As a by-product
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Isoperimetric inequalities for real-valued functions with applications to monotonicity testing Random Struct. Algorithms (IF 1.0) Pub Date : 2024-02-29 Hadley Black, Iden Kalemaj, Sofya Raskhodnikova
We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra (SICOMP 2018) for Boolean functions to the case of real-valued functions f:{0,1}d→ℝ$$ f:{\left\{0,1\right\}}^d\to \mathbb{R} $$. Our main tool in the proof of the generalized inequality is a new Boolean decomposition that represents every real-valued function f$$ f $$ over an arbitrary partially ordered domain as a collection
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Weakly saturated random graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2024-02-19 Zsolt Bartha, Brett Kolesnik
As introduced by Bollobás, a graph G$$ G $$ is weakly H$$ H $$-saturated if the complete graph Kn$$ {K}_n $$ is obtained by iteratively completing copies of H$$ H $$ minus an edge. For all graphs H$$ H $$, we obtain an asymptotic lower bound for the critical threshold pc$$ {p}_c $$, at which point the Erdős–Rényi graph 𝒢n,p is likely to be weakly H$$ H $$-saturated. We also prove an upper bound for
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A robust Corrádi–Hajnal theorem Random Struct. Algorithms (IF 1.0) Pub Date : 2024-02-09 Peter Allen, Julia Böttcher, Jan Corsten, Ewan Davies, Matthew Jenssen, Patrick Morris, Barnaby Roberts, Jozef Skokan
For a graph G$$ G $$ and p∈[0,1]$$ p\in \left[0,1\right] $$, we denote by Gp$$ {G}_p $$ the random sparsification of G$$ G $$ obtained by keeping each edge of G$$ G $$ independently, with probability p$$ p $$. We show that there exists a C>0$$ C>0 $$ such that if p≥C(logn)1/3n−2/3$$ p\ge C{\left(\log n\right)}^{1/3}{n}^{-2/3} $$ and G$$ G $$ is an n$$ n $$-vertex graph with n∈3ℕ$$ n\in 3\mathbb{N}
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On pattern-avoiding permutons Random Struct. Algorithms (IF 1.0) Pub Date : 2024-01-29 Frederik Garbe, Jan Hladký, Gábor Kun, Kristýna Pekárková
The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order k $$ k $$ have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x-axis) consists only of atoms, at most ( k − 1 ) $$ \left(k-1\right) $$ many,
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Coupling Bertoin's and Aldous–Pitman's representations of the additive coalescent Random Struct. Algorithms (IF 1.0) Pub Date : 2024-01-11 Igor Kortchemski, Paul Thévenin
We construct a coupling between two seemingly very different constructions of the standard additive coalescent, which describes the evolution of masses merging pairwise at rates proportional to their sums. The first construction, due to Aldous and Pitman, involves the components obtained by logging the Brownian continuum random tree (CRT) by a Poissonian rain on its skeleton as time increases. The
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Cycles with many chords Random Struct. Algorithms (IF 1.0) Pub Date : 2024-01-10 Nemanja Draganić, Abhishek Methuku, David Munhá Correia, Benny Sudakov
How many edges in an n$$ n $$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any n$$ n $$-vertex graph with 2n3/2$$ 2{n}^{3/2} $$ edges contains such a cycle. We significantly improve this old bound by showing that Ω(nlog8n)$$ \Omega \left(n\kern0.2em {\log}^8n\right) $$
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Greedy maximal independent sets via local limits Random Struct. Algorithms (IF 1.0) Pub Date : 2023-12-18 Michael Krivelevich, Tamás Mészáros, Peleg Michaeli, Clara Shikhelman
The random greedy algorithm for finding a maximal independent set in a graph constructs a maximal independent set by inspecting the graph's vertices in a random order, adding the current vertex to the independent set if it is not adjacent to any previously added vertex. In this paper, we present a general framework for computing the asymptotic density of the random greedy independent set for sequences
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Dixon's asymptotic without the classification of finite simple groups Random Struct. Algorithms (IF 1.0) Pub Date : 2023-12-18 Sean Eberhard
Without using the classification of finite simple groups (CFSG), we show that the probability that two random elements of Sn$$ {S}_n $$ generate a primitive group smaller than An$$ {A}_n $$ is at most exp(−c(nlogn)1/2)$$ \exp \left(-c{\left(n\log n\right)}^{1/2}\right) $$. As a corollary we get Dixon's asymptotic expansion
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Random graphs embeddable in order-dependent surfaces Random Struct. Algorithms (IF 1.0) Pub Date : 2023-12-05 Colin McDiarmid, Sophia Saller
Given a ‘genus function’ g = g ( n ) $$ g=g(n) $$ , we let E g $$ {\mathcal{E}}^g $$ be the class of all graphs G $$ G $$ such that if G $$ G $$ has order n $$ n $$ (i.e., has n $$ n $$ vertices) then it is embeddable in a surface of Euler genus at most g ( n ) $$ g(n) $$ . Let the random graph R n $$ {R}_n $$ be sampled uniformly from the graphs in E g $$ {\mathcal{E}}^g $$ on vertex set [ n ] = {
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Small cycle structure for words in conjugation invariant random permutations Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-29 Mohamed Slim Kammoun, Mylène Maïda
We study the cycle structure of words in several random permutations. We assume that the permutations are independent and that their distribution is conjugation invariant, with a good control on their short cycles. If, after successive cyclic simplifications, the word w $$ w $$ still contains at least two different letters, then we get a universal limiting joint law for short cycles for the word in
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On random irregular subgraphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-28 Jacob Fox, Sammy Luo, Huy Tuan Pham
Let G $$ G $$ be a d $$ d $$ -regular graph on n $$ n $$ vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model H = H ( G ) $$ H=H(G) $$ . Assign independently to each vertex v $$ v $$ of G $$ G $$ a uniform random number x ( v ) ∈ [ 0 , 1 ] $$ x(v)\in \left[0,1\right] $$ , and an edge ( u , v ) $$ \left(u,v\right) $$ of G $$ G $$ is an edge of
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Rainbow Hamilton cycles in random geometric graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-27 Alan Frieze, Xavier Pérez-Giménez
Let X 1 , X 2 , … , X n $$ {X}_1,{X}_2,\dots, {X}_n $$ be chosen independently and uniformly at random from the unit d $$ d $$ -dimensional cube [ 0 , 1 ] d $$ {\left[0,1\right]}^d $$ . Let r $$ r $$ be given and let 𝒳 = X 1 , X 2 , … , X n . The random geometric graph G = G 𝒳 , r has vertex set 𝒳 and an edge X i X j $$ {X}_i{X}_j $$ whenever ‖ X i − X j ‖ ≤ r $$ \left\Vert {X}_i-{X}_j\right\Vert
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Frozen 1-RSB structure of the symmetric Ising perceptron Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-23 Will Perkins, Changji Xu
We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1-RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution
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Wireless random-access networks with bipartite interference graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-23 Sem C. Borst, Frank den Hollander, Francesca R. Nardi, Matteo Sfragara
We consider random-access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states
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Note on down-set thresholds Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-22 Lutz Warnke
Gunby–He–Narayanan showed that the logarithmic gap predictions of Kahn–Kalai and Talagrand (proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about thresholds of up-sets do not apply to down-sets. In particular, for the down-set of triangle-free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler
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Counting orientations of random graphs with no directed k-cycles Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-07 Marcelo Campos, Maurício Collares, Guilherme Oliveira Mota
For every k ⩾ 3 $$ k\geqslant 3 $$ , we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length k $$ k $$ . This solves a conjecture of Kohayakawa, Morris and the last two authors.
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The number of descendants in a random directed acyclic graph Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-07 Svante Janson
We consider a well-known model of random directed acyclic graphs of order n $$ n $$ , obtained by recursively adding vertices, where each new vertex has a fixed outdegree d ⩾ 2 $$ d\geqslant 2 $$ and the endpoints of the d $$ d $$ edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number X ( n ) $$ {X}^{(n)} $$ of vertices that are descendants
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Prominent examples of flip processes Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-07 Pedro Araújo, Jan Hladký, Eng Keat Hng, Matas Šileikis
Flip processes, introduced in [Garbe, Hladký, Šileikis, Skerman: From flip processes to dynamical systems on graphons], are a class of random graph processes defined using a rule which is just a function ℛ : ℋ k → ℋ k $$ \mathcal{R}:{\mathscr{H}}_k\to {\mathscr{H}}_k $$ from all labelled graphs of a fixed order k $$ k $$ into itself. The process starts with an arbitrary given n $$ n $$ -vertex graph
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Sharp thresholds in adaptive random graph processes Random Struct. Algorithms (IF 1.0) Pub Date : 2023-11-07 Calum MacRury, Erlang Surya
The 𝒟 -process is a single player game in which the player is initially presented the empty graph on n $$ n $$ vertices. In each step, a subset of edges X $$ X $$ is independently sampled according to a distribution 𝒟 . The player then selects one edge e $$ e $$ from X $$ X $$ , and adds e $$ e $$ to its current graph. For a fixed monotone increasing graph property 𝒫 , the objective of the player
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Defective coloring of hypergraphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-27 António Girão, Freddie Illingworth, Alex Scott, David R. Wood
We prove that the vertices of every ( r + 1 ) $$ \left(r+1\right) $$ -uniform hypergraph with maximum degree Δ $$ \Delta $$ may be colored with c Δ d + 1 1 / r $$ c{\left(\frac{\Delta}{d+1}\right)}^{1/r} $$ colors such that each vertex is in at most d $$ d $$ monochromatic edges. This result, which is best possible up to the value of the constant c $$ c $$ , generalizes the classical result of Erdős
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Coloring lines and Delaunay graphs with respect to boxes Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-26 István Tomon
The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties. (i) First, we construct a family of n $$ n $$ lines in ℝ 3 $$ {\mathbb{R}}^3 $$ whose intersection graph is triangle-free of chromatic number Ω ( n 1 / 15 ) $$ \Omega \left({n}^{1/15}\right) $$ . This improves the previously best
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Rainbow subdivisions of cliques Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-25 Tao Jiang, Shoham Letzter, Abhishek Methuku, Liana Yepremyan
We show that for every integer m ≥ 2 $$ m\ge 2 $$ and large n $$ n $$ , every properly edge-colored graph on n $$ n $$ vertices with at least n ( log n ) 53 $$ n{\left(\log n\right)}^{53} $$ edges contains a rainbow subdivision of K m $$ {K}_m $$ . This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs.
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The Erlang weighted tree, a new branching process Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-23 M. Moharrami, V. Subramanian, M. Liu, R. Sundaresan
In this paper, we study a new discrete tree and the resulting branching process, which we call the erlang weighted tree(EWT). The EWT appears as the local weak limit of a random graph model proposed in La and Kabkab, Internet Math. 11 (2015), no. 6, 528–554. In contrast to the local weak limit of well-known random graph models, the EWT has an interdependent structure. In particular, its vertices encode
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Spatial mixing and the random-cluster dynamics on lattices Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-20 Reza Gheissari, Alistair Sinclair
An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in particular, the classical notions of weak and strong spatial mixing, which have been used to show the best known mixing time bounds in the high-temperature regime for the
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On a rainbow extremal problem for color-critical graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-17 Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, Hong Liu, Jaehyeon Seo
Given k$$ k $$ graphs G1,…,Gk$$ {G}_1,\dots, {G}_k $$ over a common vertex set of size n$$ n $$, what is the maximum value of ∑i∈[k]e(Gi)$$ {\sum}_{i\in \left[k\right]}e\left({G}_i\right) $$ having no “colorful” copy of H$$ H $$, that is, a copy of H$$ H $$ containing at most one edge from each Gi$$ {G}_i $$? Keevash, Saks, Sudakov, and Verstraëte denoted this number as exk(n,H)$$ {\mathrm{ex}}_k\left(n
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The upper tail problem for induced 4-cycles in sparse random graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-16 Asaf Cohen Antonir
Building on the techniques from the breakthrough paper of Harel, Mousset and Samotij, which solved the upper tail problem for cliques, we compute the asymptotics of the upper tail for the number of induced copies of the 4-cycle in the binomial random graph G n , p $$ {G}_{n,p} $$ . We observe a new phenomenon in the theory of large deviations of subgraph counts. This phenomenon is that, in a certain
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A down-up chain with persistent labels on multifurcating trees Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-12 Frederik Sørensen
In this article, we propose to study a general notion of a down-up Markov chain for multifurcating trees with n $$ n $$ labeled leaves. We study in detail down-up chains associated with the ( α , γ ) $$ \left(\alpha, \gamma \right) $$ -model of Chen et al. (Electron. J. Probab. 14 (2009), 400–430.), generalizing and further developing previous work by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804
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A full characterization of invariant embeddability of unimodular planar graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-10-04 Ádám Timár, László Márton Tóth
When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in Benjamini and Timar, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends
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Expansion of random 0/1 polytopes Random Struct. Algorithms (IF 1.0) Pub Date : 2023-08-31 Brett Leroux, Luis Rademacher
A conjecture of Milena Mihail and Umesh Vazirani (Proc. 24th Annu. ACM Symp. Theory Comput., ACM, Victoria, BC, 1992, pp. 26–38.) states that the edge expansion of the graph of every 0 / 1 $$ 0/1 $$ polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be
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Extremal results on feedback arc sets in digraphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-08-21 Jacob Fox, Zoe Himwich, Nitya Mani
For an oriented graph G$$ G $$, let β(G)$$ \beta (G) $$ denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any m$$ m $$-edge oriented graph G$$ G $$ satisfies β(G)=m/2−Ω(m3/4)$$ \beta (G)=m/2-\Omega \left({m}^{3/4}\right) $$. We observe that if an oriented graph G$$ G $$ has a fixed forbidden subgraph B$$ B
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Universality of superconcentration in the Sherrington–Kirkpatrick model Random Struct. Algorithms (IF 1.0) Pub Date : 2023-08-11 Wei-Kuo Chen, Wai-Kit Lam
We study the universality of superconcentration for the free energy in the Sherrington–Kirkpatrick model. In [10], Chatterjee showed that when the system consists of N $$ N $$ spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order N / log N $$ N/\log N $$ , in contrast to the O ( N ) $$ O(N) $$ bound obtained from the Gaussian–Poincaré
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The birth of the strong components Random Struct. Algorithms (IF 1.0) Pub Date : 2023-08-07 Sergey Dovgal, Élie de Panafieu, Dimbinaina Ralaivaosaona, Vonjy Rasendrahasina, Stephan Wagner
It is known that random directed graphs D(n,p)$$ D\left(n,p\right) $$ undergo a phase transition around the point p=1/n$$ p=1/n $$. Earlier, Łuczak and Seierstad have established that as n→∞$$ n\to \infty $$ when p=(1+μn−1/3)/n$$ p=\left(1+\mu {n}^{-1/3}\right)/n $$, the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases
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A lower bound for set-coloring Ramsey numbers Random Struct. Algorithms (IF 1.0) Pub Date : 2023-08-03 Lucas Aragão, Maurício Collares, João Pedro Marciano, Taísa Martins, Robert Morris
The set-coloring Ramsey number R r , s ( k ) $$ {R}_{r,s}(k) $$ is defined to be the minimum n $$ n $$ such that if each edge of the complete graph K n $$ {K}_n $$ is assigned a set of s $$ s $$ colors from { 1 , … , r } $$ \left\{1,\dots, r\right\} $$ , then one of the colors contains a monochromatic clique of size k $$ k $$ . The case s = 1 $$ s=1 $$ is the usual r $$ r $$ -color Ramsey number, and
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Optimal bisections of directed graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-08-02 Guanwu Liu, Jie Ma, Chunlei Zu
In this article, motivated by a problem of Scott [Surveys in combinatorics, 327 (2005), 95-117.] and a conjecture of Lee et al. [Random Struct. Algorithm, 48 (2016), 147-170.] we consider bisections of directed graphs. We prove that every directed graph with m $$ m $$ arcs and minimum semidegree at least d $$ d $$ admits a bisection in which at least d 2 ( 2 d + 1 ) + o ( 1 ) m $$ \left(\frac{d}{2
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Threshold for detecting high dimensional geometry in anisotropic random geometric graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-08-01 Matthew Brennan, Guy Bresler, Brice Huang
In the anisotropic random geometric graph model, vertices correspond to points drawn from a high-dimensional Gaussian distribution and two vertices are connected if their distance is smaller than a specified threshold. We study when it is possible to hypothesis test between such a graph and an Erdős-Rényi graph with the same edge probability. If n $$ n $$ is the number of vertices and α $$ \alpha $$
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Packing list-colorings Random Struct. Algorithms (IF 1.0) Pub Date : 2023-07-31 Stijn Cambie, Wouter Cames van Batenburg, Ewan Davies, Ross J. Kang
List coloring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-coloring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a k $$ k $$ -list-assignment L $$ L $$ of a graph G $$ G $$ , which is the assignment of
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Subcritical monotone cellular automata Random Struct. Algorithms (IF 1.0) Pub Date : 2023-07-31 Paul Balister, Béla Bollobás, Robert Morris, Paul Smith
We study monotone cellular automata (also known as 𝒰 -bootstrap percolation) in ℤ d $$ {\mathbb{Z}}^d $$ with random initial configurations. Confirming a conjecture of Balister, Bollobás, Przykucki and Smith, who proved the corresponding result in two dimensions, we show that the critical probability is non-zero for all subcritical models.
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A Ramsey–Turán theory for tilings in graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-07-31 Jie Han, Patrick Morris, Guanghui Wang, Donglei Yang
For a k $$ k $$ -vertex graph F $$ F $$ and an n $$ n $$ -vertex graph G $$ G $$ , an F $$ F $$ -tiling in G $$ G $$ is a collection of vertex-disjoint copies of F $$ F $$ in G $$ G $$ . For r ∈ ℕ $$ r\in \mathbb{N} $$ , the r $$ r $$ -independence number of G $$ G $$ , denoted α r ( G ) $$ {\alpha}_r(G) $$ , is the largest size of a K r $$ {K}_r $$ -free set of vertices in G $$ G $$ . In this article
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Limit theorems for patterns in ranked tree-child networks Random Struct. Algorithms (IF 1.0) Pub Date : 2023-07-29 Michael Fuchs, Hexuan Liu, Tsan-Cheng Yu
We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree-child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (Random Struct. Algoritm. 60 (2022), no. 4, 653–689). For patterns of height 1 and 2, we show that they either occur frequently (mean is asymptotically linear and limit law is normal)
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Random perfect matchings in regular graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-07-12 Bertille Granet, Felix Joos
We prove that in all regular robust expanders G $$ G $$ , every edge is asymptotically equally likely contained in a uniformly chosen perfect matching M $$ M $$ . We also show that given any fixed matching or spanning regular graph N $$ N $$ in G $$ G $$ , the random variable | M ∩ E ( N ) | $$ \mid M\cap E(N)\mid $$ is approximately Poisson distributed. This in particular confirms a conjecture and
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A fourth-moment phenomenon for asymptotic normality of monochromatic subgraphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-06-28 Sayan Das, Zoe Himwich, Nitya Mani
Given a graph sequence { G n } n ≥ 1 $$ {\left\{{G}_n\right\}}_{n\ge 1} $$ and a simple connected subgraph H $$ H $$ , we denote by T ( H , G n ) $$ T\left(H,{G}_n\right) $$ the number of monochromatic copies of H $$ H $$ in a uniformly random vertex coloring of G n $$ {G}_n $$ with c ≥ 2 $$ c\ge 2 $$ colors. We prove a central limit theorem for T ( H , G n ) $$ T\left(H,{G}_n\right) $$ (we denote
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The impact of heterogeneity and geometry on the proof complexity of random satisfiability Random Struct. Algorithms (IF 1.0) Pub Date : 2023-06-28 Thomas Bläsius, Tobias Friedrich, Andreas Göbel, Jordi Levy, Ralf Rothenberger
Satisfiability is considered the canonical NP-complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic
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Concentration estimates for functions of finite high-dimensional random arrays Random Struct. Algorithms (IF 1.0) Pub Date : 2023-06-27 Pandelis Dodos, Konstantinos Tyros, Petros Valettas
Let X $$ \boldsymbol{X} $$ be a d $$ d $$ -dimensional random array on [ n ] $$ \left[n\right] $$ whose entries take values in a finite set 𝒳 , that is, X = ⟨ X s : s ∈ [ n ] d ⟩ $$ \boldsymbol{X}=\left\langle {X}_s:s\in \left(\genfrac{}{}{0ex}{}{\left[n\right]}{d}\right)\right\rangle $$ is an 𝒳 -valued stochastic process indexed by the set [ n ] d $$ \left(\genfrac{}{}{0ex}{}{\left[n\right]}{d}\right)
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Reversible random walks on dynamic graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-06-21 Nobutaka Shimizu, Takeharu Shiraga
This paper discusses random walks on edge-changing dynamic graphs. We prove general and improved bounds for mixing, hitting, and cover times for a random walk according to a sequence of irreducible and reversible transition matrices with the time-invariant stationary distribution. An interesting consequence is the tight bounds of the lazy Metropolis walk on any dynamic connected graph. We also prove
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Cycle lengths in randomly perturbed graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-06-20 Elad Aigner-Horev, Dan Hefetz, Michael Krivelevich
Let G $$ G $$ be an n $$ n $$ -vertex graph, where δ ( G ) ≥ δ n $$ \delta (G)\ge \delta n $$ for some δ : = δ ( n ) $$ \delta := \delta (n) $$ . A result of Bohman, Frieze and Martin from 2003 asserts that if α ( G ) = O δ 2 n $$ \alpha (G)=O\left({\delta}^2n\right) $$ , then perturbing G $$ G $$ via the addition of ω log ( 1 / δ ) δ 3 $$ \omega \left(\frac{\log \left(1/\delta \right)}{\delta^3}\right)
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Growing uniform planar maps face by face Random Struct. Algorithms (IF 1.0) Pub Date : 2023-06-19 Alessandra Caraceni, Alexandre Stauffer
We provide “growth schemes” for inductively generating uniform random 2 p $$ 2p $$ -angulations of the sphere with n $$ n $$ faces, as well as uniform random simple triangulations of the sphere with 2 n $$ 2n $$ faces. In the case of 2 p $$ 2p $$ -angulations, we provide a way to insert a new face at a random location in a uniform 2 p $$ 2p $$ -angulation with n $$ n $$ faces in such a way that the
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Cycles in Mallows random permutations Random Struct. Algorithms (IF 1.0) Pub Date : 2023-06-19 Jimmy He, Tobias Müller, Teun W. Verstraaten
We study cycle counts in permutations of 1 , … , n $$ 1,\dots, n $$ drawn at random according to the Mallows distribution. Under this distribution, each permutation π ∈ S n $$ \pi \in {S}_n $$ is selected with probability proportional to q inv ( π ) $$ {q}^{\mathrm{inv}\left(\pi \right)} $$ , where q > 0 $$ q>0 $$ is a parameter and inv ( π ) $$ \mathrm{inv}\left(\pi \right) $$ denotes the number of
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Matchings on trees and the adjacency matrix: A determinantal viewpoint Random Struct. Algorithms (IF 1.0) Pub Date : 2023-06-15 András Mészáros
Let G $$ G $$ be a finite tree. For any matching M $$ M $$ of G $$ G $$ , let U ( M ) $$ U(M) $$ be the set of vertices uncovered by M $$ M $$ . Let ℳ G $$ {\mathcal{M}}_G $$ be a uniform random maximum size matching of G $$ G $$ . In this paper, we analyze the structure of U ( ℳ G ) $$ U\left({\mathcal{M}}_G\right) $$ . We first show that U ( ℳ G ) $$ U\left({\mathcal{M}}_G\right) $$ is a determinantal
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Hypergraph Ramsey numbers of cliques versus stars Random Struct. Algorithms (IF 1.0) Pub Date : 2023-05-18 David Conlon, Jacob Fox, Xiaoyu He, Dhruv Mubayi, Andrew Suk, Jacques Verstraëte
Let K m ( 3 ) $$ {K}_m^{(3)} $$ denote the complete 3-uniform hypergraph on m $$ m $$ vertices and S n ( 3 ) $$ {S}_n^{(3)} $$ the 3-uniform hypergraph on n + 1 $$ n+1 $$ vertices consisting of all n 2 $$ \left(\genfrac{}{}{0ex}{}{n}{2}\right) $$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal
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Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree Random Struct. Algorithms (IF 1.0) Pub Date : 2023-05-10 Ahmed El Alaoui, Andrea Montanari, Mark Sellke
Given a graph G $$ G $$ of degree k $$ k $$ over n $$ n $$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth 2 L $$ 2L $$ , we develop a local message passing algorithm whose complexity is O ( n k L ) $$ O(nkL) $$ , and that achieves near optimal cut values among all L $$ L $$ -local algorithms. Focusing on max-cut
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Natural quasirandomness properties Random Struct. Algorithms (IF 1.0) Pub Date : 2023-05-03 Leonardo N. Coregliano, Alexander A. Razborov
The theory of quasirandomness has greatly expanded from its inaugural graph theoretical setting to several different combinatorial objects such as hypergraphs, tournaments, permutations and so forth. However, these quasirandomness variants have been done in an ad-hoc case-by-case manner. In this article, we propose three new hierarchies of quasirandomness properties that can be naturally defined for
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Independence number of hypergraphs under degree conditions Random Struct. Algorithms (IF 1.0) Pub Date : 2023-04-22 Vojtěch Rödl, Marcelo Sales, Yi Zhao
A well-known result of Ajtai Komlós, Pintz, Spencer, and Szemerédi (J. Combin. Theory Ser. A 32 (1982), 321–335) states that every k $$ k $$ -graph H $$ H $$ on n $$ n $$ vertices, with girth at least five, and average degree t k − 1 $$ {t}^{k-1} $$ contains an independent set of size c n ( log t ) 1 / ( k − 1 ) / t $$ cn{\left(\log t\right)}^{1/\left(k-1\right)}/t $$ for some c > 0 $$ c>0 $$ . In
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On powers of tight Hamilton cycles in randomly perturbed hypergraphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-04-17 Yulin Chang, Jie Han, Lubos Thoma
For integers k ≥ 3 $$ k\ge 3 $$ and r ≥ 2 $$ r\ge 2 $$ , we show that for every α > 0 $$ \alpha >0 $$ , there exists ε > 0 $$ \varepsilon >0 $$ such that the union of k $$ k $$ -uniform hypergraph on n $$ n $$ vertices with minimum codegree at least α n $$ \alpha n $$ and a binomial random k $$ k $$ -uniform hypergraph G ( k ) ( n , p ) $$ {G}^{(k)}\left(n,p\right) $$ with p ≥ n − k + r − 2 k − 1 −
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Moderate deviations in cycle count Random Struct. Algorithms (IF 1.0) Pub Date : 2023-04-17 Joe Neeman, Charles Radin, Lorenzo Sadun
We prove moderate deviations bounds for the lower tail of the number of odd cycles in a 𝒢 ( n , m ) random graph. We show that the probability of decreasing triangle density by t 3 , is exp ( − Θ ( n 2 t 2 ) ) whenever n − 3 / 4 ≪ t 3 ≪ 1 . These complement results of Goldschmidt, Griffiths, and Scott, who showed that for n − 3 / 2 ≪ t 3 ≪ n − 1 , the probability is exp ( − Θ ( n 3 t 6 ) ) . That
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Deterministic algorithms for the Lovász local lemma: Simpler, more general, and more parallel Random Struct. Algorithms (IF 1.0) Pub Date : 2023-04-15 David G. Harris
The Lovász local lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection ℬ $$ \mathcal{B} $$ of “bad” events which are mostly independent and have low probability. A seminal algorithm of Moser and Tardos (J. ACM, 2010, 57, 11) (which we call the MT algorithm) gives nearly-automatic randomized algorithms for most constructions
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Down-set thresholds Random Struct. Algorithms (IF 1.0) Pub Date : 2023-04-12 Benjamin Gunby, Xiaoyu He, Bhargav Narayanan
We elucidate the relationship between the threshold and the expectation-threshold of a down-set. Qualitatively, our main result demonstrates that there exist down-sets with polynomial gaps between their thresholds and expectation-thresholds; in particular, the logarithmic gap predictions of Kahn–Kalai and Talagrand (recently proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about up-sets do not
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The spectral gap of random regular graphs Random Struct. Algorithms (IF 1.0) Pub Date : 2023-04-11 Amir Sarid
We bound the second eigenvalue of random d $$ d $$ -regular graphs, for a wide range of degrees d $$ d $$ , using a novel approach based on Fourier analysis. Let G n , d $$ {G}_{n,d} $$ be a uniform random d $$ d $$ -regular graph on n $$ n $$ vertices, and λ ( G n , d ) $$ \lambda \left({G}_{n,d}\right) $$ be its second largest eigenvalue by absolute value. For some constant c > 0 $$ c>0 $$ and any
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Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees Random Struct. Algorithms (IF 1.0) Pub Date : 2023-03-15 Günter Last, Giovanni Peccati, D. Yogeshwaran
Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier-Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for
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Hypergraph regularity and random sampling Random Struct. Algorithms (IF 1.0) Pub Date : 2023-03-01 Felix Joos, Jaehoon Kim, Daniela Kühn, Deryk Osthus
Suppose that a k -uniform hypergraph H satisfies a certain regularity instance (that is, there is a partition of H given by the hypergraph regularity lemma into a bounded number of quasirandom subhypergraphs of prescribed densities). We prove that with high probability a large enough uniform random sample of the vertex set of H also admits the same regularity instance. Here the crucial feature is that