Let be the largest size of an induced -free subgraph that every -vertex -free graph is guaranteed to contain. We prove that for any triangle-free graph , Along the way we give a slight improvement of a construction of Erd & odblac;s-Frankl-R & ouml;dl for the Brown-Erd & odblac;s-S & oacute;s -problem when is large. In contrast to our result for , for any -free graph containing a cycle, we prove there exists such that For every graph , we prove that there exists such that whenever is a non-empty graph such that is not contained in any blowup of , then . On the other hand, for graph that is not a clique, and every , we exhibit a -free graph such that .
ABSTRACT We study the chromatic number of typical triangle‐free graphs with edges and establish the width of the scaling window for the transitions from to and from to . The transition from 3‐ to 4‐colorability has scaling window of width . To prove this, we show a high probability equivalence of the 3‐colorability of a random triangle‐free graph at this density and the satisfiability of an instance of bipartite random 2‐SAT, for which we establish the width of the scaling window following the techniques of Bollobás, Borgs, Chayes, Kim, and Wilson. The transition from 4‐ to 5‐colorability has scaling window of width . To prove this, we show a high probability equivalence of the 4‐colorability of a random triangle‐free graph at this density and the simultaneous 2‐colorability of two independent Erdős–Rényi random graphs. For this transition, we also establish the limiting probability of 4‐colorability inside the scaling window.
We study the inference of the root for a random nearest neighbor tree generated by sequentially embedding vertices uniformly at random in the d-dimensional torus and connecting each new vertex to the nearest existing vertex. Given an error parameter and the unlabeled tree, we want to efficiently find a small "confidence set" of epsilon > 0 candidate vertices containing the root with probability at least 1-epsilon. We define several problem variations-such as embedded and metric root finding-which differ based on the available metric information provided in addition to the graph structure (torus embedding, edge lengths, or none). For embedded and metric root finding, we construct efficient algorithms and derive bounds on the confidence set size. For embedded root finding, the upper bound is subpolynomial in 1/epsilon and the information-theoretic lower bound is polylogarithmic in 1/epsilon . For embedded root finding in d=1 , we obtain matching upper and lower bounds for a confidence set of size Theta (log(1/epsilon)/log log(1/epsilon)) .
We provide new exact formulas for the distribution of massive spanning forests on the complete graph, which give also a new outlook on the celebrated special case of the uniform spanning tree. As a corollary we identify their local limit. This generalizes a well-known theorem of Grimmett on the local limit of uniform spanning trees on the complete graph.
We study a random walk on the Lie algebra sl(2)(F-p) where new elements are produced by randomly applying adjoint operators of two generators. Focusing on the generic case where the generators are selected at random, we analyze the limiting distribution of the random walk and the speed at which it converges to this distribution. These questions reduce to the study of a random walk on a cyclic group. We show that, with high probability, the walk exhibits a pre-cutoff phenomenon after roughly p steps. Notably, the limiting distribution need not be uniform, and it depends on the prime divisors of p-1. Furthermore, we prove that by incorporating a simple random twist into the walk, we can embed a well-known affine random walk on F-p into the modified random Lie bracket, allowing us to show that the entire Lie algebra is covered in roughly p steps in the generic case.
In this paper, we study the problem of finding a collection of planted cycles in an Erd & ouml;s-R & eacute;nyi random graph G similar to & Gscr; ( n , lambda / n ) , in analogy to the famous Planted Clique Problem. When the cycles are planted on a uniformly random subset of vertices, we show that almost-exact recovery (i.e., recovering all but a vanishing fraction of planted-cycle edges as ) is information-theoretically possible if and impossible if . Moreover, despite the worst-case computational hardness of finding long cycles, we design a polynomial-time algorithm that attains almost exact recovery when . This stands in stark contrast to the Planted Clique Problem, where a significant computational-statistical gap is widely conjectured. [A key technical contribution is a novel generating-function approach for counting imbalanced circuits that arise in decompositions of the symmetric difference between the planted cycles and alternative feasible solutions.]
We study the contact process on the long-range percolation cluster on & Zopf; where each edge < i, j > is open with probability|i-j|(-s) fors>2. Using a renormalization procedure, we apply the Peierls-type argument to prove that the contact process dies out if the transmission rate is lower than a critical threshold. Our methods involve the control of crossing probabilities for percolation on randomly-stretched lattices as in Hilario et al
ABSTRACT A graph is even‐degenerate if one can iteratively remove a vertex of even degree at each step until at most one edge remains. Recently, Janzer and Yip showed that the Erdős–Rényi random graph is even‐degenerate with high probability, and asked whether an analogous result holds for any general . In this paper, we answer this question for any constant in affirmation by proving that is even‐degenerate with high probability.
The discrepancy between two independent samples \(X_1,\dots,X_n\) and \(Y_1,\dots,Y_n\) drawn from the same distribution on $\mathbb{R}^d$ typically has order \(O(\sqrt{n})\) even in one dimension. We give a simple online algorithm that reduces the discrepancy to \(O(\log^{2d} n)\) by discarding a small fraction of the points.
ABSTRACT Binary search trees (BSTs) are fundamental data structures whose performance is largely governed by tree height. We introduce a block model for constructing BSTs by embedding internal BSTs into the nodes of an external BST—a structure motivated by parallel data architectures—corresponding to composite permutations formed via Kronecker or wreath products. Extending Devroye's result that the height of a random BST satisfies , we show that block BSTs with nodes and fixed external size satisfy in distribution. We then study butterfly trees : BSTs with nodes generated from permutations built using iterated Kronecker or wreath products. For simple butterfly trees (from iterated Kronecker products of ), we give a full distributional description showing polynomial height growth: with . For nonsimple butterfly trees (from wreath products), we prove power‐law bounds: , with .
ABSTRACT Given a graph , the percolated graph is formed by retaining each edge independently with probability . Collares, Diskin, Erde, and Krivelevich initiated the study of large structures in percolated single‐scale vertex‐expander graphs, wherein every set of exactly vertices of has at least neighbors before percolation. We extend their result to a conjectured stronger form, proving that if and is a graph on at least vertices which expands as above, then contains a cycle of length with probability at least as .
ABSTRACT A proper ‐coloring of a graph is called ‐conflict‐free if every vertex has at least colors appearing exactly once in its neighborhood. Let denote the minimum such that such a coloring exists. We show that for every fixed , every graph of maximum degree satisfies . This expands on the work of Cho et al., and improves a recent result of Liu and Reed in the case . We conjecture that for every and every graph of maximum degree sufficiently large, the bound should hold, which would be tight. When the minimum degree of is sufficiently large, namely , we show that this upper bound can be further reduced to . This improves a recent bound from Kamyczura and Przybyło when .
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.
ABSTRACT Many important results in extremal graph theory can be roughly summarized as “if a triangle‐free graph has certain properties, then it has a homomorphism to a triangle‐free graph of bounded size.” For example, bounds on homomorphism thresholds give such a statement if has sufficiently high minimum degree, and the approximate homomorphism theorem gives such a statement for all if one weakens the notion of homomorphism appropriately. In this paper, we study asymmetric versions of these results, where the assumptions on and need not match. For example, we prove that if is a graph with odd girth at least 9 and minimum degree at least , then is homomorphic to a triangle‐free graph whose size depends only on . Moreover, the odd girth assumption can be weakened to odd girth at least 7 if has bounded VC dimension or bounded domination number. This gives a new and improved proof of a result of Huang, Liu, Rong, and Xu. We also prove that in the asymmetric approximate homomorphism theorem, the bounds exhibit a rather surprising “double phase transition”: the bounds are super‐exponential if is only assumed to be triangle‐free, they become exponential if is assumed to have odd girth 7 or 9, and become linear if has odd girth at least 11. Our proofs use a wide variety of techniques, including entropy arguments, the Frieze–Kannan weak regularity lemma, properties of the generalized Mycielskian construction, and recent work on abundance and the asymmetric removal lemma.
Locally Markov walks are natural generalizations of classical Markov chains, where instead of a particle moving independently of the past, it decides where to move next depending on the last action performed at the current location. We introduce the concept of locally Markov walks and we describe their stationary distribution and recurrent states, and we prove several properties such as irreducibility and ergodicity. For a particular locally Markov walk - the uniform unicycle walk on the complete graph - we investigate the mixing time and we prove that it exhibits cutoff.
We construct a unimodular random rooted graph with maximal degree d≥ 3 and upper growth rate d-1, which does not have a growth rate. Abért, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least √(d-1), then the growth rate exists, and asked with some scepticism if this may hold for more general graphs. Our construction shows that the answer is negative. We also provide a non-hyperfinite example of a unimodular random graph with no growth rate. This may be of interest in light of a conjecture of Abért that unimodular Riemannian surfaces of bounded negative curvature always have growth.
We establish a refined version of a graph container lemma due to Galvin and discuss several applications related to the hard-core model on bipartite expander graphs. Given a graph G and lambda> 0, the hard-core model on G at activity lambda is the probability distribution mu(G lambda )independent sets in G given by mu(G lambda)(I) proportional to lambda |I| . As one of our main applications, we show that the hard-core model at activity on lambda the hypercube Q(d ) exhibits a 'structured phase' for lambda= Omega(log(2)d/d(1/2)) in the following sense: in a typical sample from mu(QG lambda), most vertices are contained in one side of the bipartition of Q(d). This improves upon a result of Galvin, which establishes the same for lambda= Omega(logd/d(1/3)). As another application, we establish a fully polynomial-time approximation scheme (FPTAS) for the hard-core model on a d-regular bipartite establish a fully polynomial-time approximation scheme (FPTAS) for the hard-core model on a d-regular bipartite alpha-expander, with alpha> 0 fixed, when lambda= Omega(log(2)d/d(1/2)). This improves upon the bound lambda= Omega(logd/d(1/4)) due to the first author, Perkins and Potukuchi. We discuss similar improvements to results of Galvin-Tetali, Balogh-Garcia-Li and Kronenberg-Spinka.
A theorem of Shearer states that every n-vertex triangle-free graph of maximum degree d >= 2 contains an independent set of size at least (d log d- d+ 1)/(d - 1)(2) center dot n. Ajtai, Koml & oacute;s, Pintz, Spencer and Szemer & eacute;di proved that every (r + 1)-uniform n-vertex "uncrowded" hypergraph of maximum degree d >= 1 has an independent set of size at least c(r)(log d)(1/r)/d(1/r ) center dot n for some c(r )> 0 depending only on r. Shearer asked whether his method for triangle-free graphs could be extended to uniform hypergraphs. In this paper, we answer this in the affirmative, thereby giving a short proof of the theorem of Ajtai, Koml & oacute;s, Pintz, Spencer and Szemer & eacute;di for a wider class of "locally sparse" hypergraphs.
Asymptotic behaviour of maximum sizes of induced trees and forests has been studied extensively in last decades, though the overall picture is far from being complete. In this paper, we close several significant gaps: 1) We prove 2-point concentration of the maximum sizes of an induced forest and an induced tree with maximum degree at most Δ in dense binomial random graphs G(n,p) with constant probability p. 2) We show concentration in an explicit interval of size o(1/p) for the maximum size of an induced forest with maximum degree at most Δ for 1/n≪ p=o(1). Our proofs rely on both the second moment approach, with the probabilistic part involving Talagrand's concentration inequality and the analytical part involving saddle-point analysis, and new results on enumeration of labelled trees and forests that might be of their own interest.
