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Antidirected subgraphs of oriented graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2024-03-06 Maya Stein, Camila Zárate-Guerén
We show that for every $\eta \gt 0$ every sufficiently large $n$ -vertex oriented graph $D$ of minimum semidegree exceeding $(1+\eta )\frac k2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k\ge \eta n$ . In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs. Further, we show that in the
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Large monochromatic components in expansive hypergraphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2024-03-05 Deepak Bal, Louis DeBiasio
A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$-colouring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$. We prove a result which says that if one replaces $K_n^k$ in Gyárfás’ theorem by any ‘expansive’ $k$-uniform hypergraph on $n$ vertices (that is, a $k$-uniform hypergraph $G$ on $n$ vertices in which
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A class of graphs of zero Turán density in a hypercube Comb. Probab. Comput. (IF 0.9) Pub Date : 2024-03-05 Maria Axenovich
For a graph $H$ and a hypercube $Q_n$ , $\textrm{ex}(Q_n, H)$ is the largest number of edges in an $H$ -free subgraph of $Q_n$ . If $\lim _{n \rightarrow \infty } \textrm{ex}(Q_n, H)/|E(Q_n)| \gt 0$ , $H$ is said to have a positive Turán density in a hypercube or simply a positive Turán density; otherwise, it has zero Turán density. Determining $\textrm{ex}(Q_n, H)$ and even identifying whether $H$
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Sharp bounds for a discrete John’s theorem Comb. Probab. Comput. (IF 0.9) Pub Date : 2024-03-05 Peter van Hintum, Peter Keevash
Tao and Vu showed that every centrally symmetric convex progression $C\subset \mathbb{Z}^d$ is contained in a generalized arithmetic progression of size $d^{O(d^2)} \# C$ . Berg and Henk improved the size bound to $d^{O(d\log d)} \# C$ . We obtain the bound $d^{O(d)} \# C$ , which is sharp up to the implied constant and is of the same form as the bound in the continuous setting given by John’s theorem
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Small subsets with large sumset: Beyond the Cauchy–Davenport bound Comb. Probab. Comput. (IF 0.9) Pub Date : 2024-02-21 Jacob Fox, Sammy Luo, Huy Tuan Pham, Yunkun Zhou
For a subset $A$ of an abelian group $G$ , given its size $|A|$ , its doubling $\kappa =|A+A|/|A|$ , and a parameter $s$ which is small compared to $|A|$ , we study the size of the largest sumset $A+A'$ that can be guaranteed for a subset $A'$ of $A$ of size at most $s$ . We show that a subset $A'\subseteq A$ of size at most $s$ can be found so that $|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$
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On a conjecture of Conlon, Fox, and Wigderson Comb. Probab. Comput. (IF 0.9) Pub Date : 2024-02-16 Chunchao Fan, Qizhong Lin, Yuanhui Yan
For graphs $G$ and $H$ , the Ramsey number $r(G,H)$ is the smallest positive integer $N$ such that any red/blue edge colouring of the complete graph $K_N$ contains either a red $G$ or a blue $H$ . A book $B_n$ is a graph consisting of $n$ triangles all sharing a common edge. Recently, Conlon, Fox, and Wigderson conjectured that for any $0\lt \alpha \lt 1$ , the random lower bound $r(B_{\lceil \alpha
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Percolation on irregular high-dimensional product graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-12-20 Sahar Diskin, Joshua Erde, Mihyun Kang, Michael Krivelevich
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 (J. Graph Theory, 99(4):651–670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the
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Product structure of graph classes with bounded treewidth Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-12-07 Rutger Campbell, Katie Clinch, Marc Distel, J. Pascal Gollin, Kevin Hendrey, Robert Hickingbotham, Tony Huynh, Freddie Illingworth, Youri Tamitegama, Jane Tan, David R. Wood
We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$ , for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H)
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On minimum spanning trees for random Euclidean bipartite graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-30 Mario Correddu, Dario Trevisan
We consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$ -th power of their Euclidean distance, with $p\gt 0$ . In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$ , we show that the maximum vertex
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Threshold graphs maximise homomorphism densities Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-29 Grigoriy Blekherman, Shyamal Patel
Given a fixed graph $H$ and a constant $c \in [0,1]$ , we can ask what graphs $G$ with edge density $c$ asymptotically maximise the homomorphism density of $H$ in $G$ . For all $H$ for which this problem has been solved, the maximum is always asymptotically attained on one of two kinds of graphs: the quasi-star or the quasi-clique. We show that for any $H$ the maximising $G$ is asymptotically a threshold
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Spread-out limit of the critical points for lattice trees and lattice animals in dimensions Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-20 Noe Kawamoto, Akira Sakai
A spread-out lattice animal is a finite connected set of edges in $\{\{x,y\}\subset \mathbb{Z}^d\;:\;0\lt \|x-y\|\le L\}$. A lattice tree is a lattice animal with no loops. The best estimate on the critical point $p_{\textrm{c}}$ so far was achieved by Penrose (J. Stat. Phys. 77, 3–15, 1994) : $p_{\textrm{c}}=1/e+O(L^{-2d/7}\log L)$ for both models for all $d\ge 1$. In this paper, we show that $p_
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The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-16 Yassine El Maazouz, Jim Pitman
The factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n \gt 0$ is the anti-derivative of $b_{n-1}(x)$ subject to $\int _0^1 b_n(x) dx = 0$. We offer a related characterization: $b_1(x) = x - 1/2$ and $({-}1)^{n-1} b_n(x)$ for $n \gt 0$ is the $n$-fold circular convolution of $b_1(x)$ with itself. Equivalently, $1 - 2^n
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Large cliques or cocliques in hypergraphs with forbidden order-size pairs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-16 Maria Axenovich, Domagoj Bradač, Lior Gishboliner, Dhruv Mubayi, Lea Weber
The well-known Erdős-Hajnal conjecture states that for any graph $F$ , there exists $\epsilon \gt 0$ such that every $n$ -vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon }$ . We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that
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Spanning trees in graphs without large bipartite holes Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-14 Jie Han, Jie Hu, Lidan Ping, Guanghui Wang, Yi Wang, Donglei Yang
We show that for any $\varepsilon \gt 0$ and $\Delta \in \mathbb{N}$ , there exists $\alpha \gt 0$ such that for sufficiently large $n$ , every $n$ -vertex graph $G$ satisfying that $\delta (G)\geq \varepsilon n$ and $e(X, Y)\gt 0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $\alpha n$ contains all spanning trees with maximum degree at most $\Delta$ . This strengthens a result
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Approximate discrete entropy monotonicity for log-concave sums Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-13 Lampros Gavalakis
It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then\begin{equation*} H(X_1+\cdots +X_{n+1}) \geq H(X_1+\cdots +X_{n}) + \frac {1}{2}\log {\Bigl (\frac {n+1}{n}\Bigr )} - o(1) \end{equation*}as $H(X_1) \to \infty$, where $H(X_1)$ denotes the (discrete)
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A special case of Vu’s conjecture: colouring nearly disjoint graphs of bounded maximum degree Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-10 Tom Kelly, Daniela Kühn, Deryk Osthus
A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every fixed $C$, if each vertex $v \in \bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$, then the (list) chromatic number of $\bigcup _{i=1}^m G_i$
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Mastermind with a linear number of queries Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-08 Anders Martinsson, Pascal Su
Since the 1960s Mastermind has been studied for the combinatorial and information-theoretical interest the game has to offer. Many results have been discovered starting with Erdős and Rényi determining the optimal number of queries needed for two colours. For $k$ colours and $n$ positions, Chvátal found asymptotically optimal bounds when $k \le n^{1-\varepsilon }$. Following a sequence of gradual improvements
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On oriented cycles in randomly perturbed digraphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-08 Igor Araujo, József Balogh, Robert A. Krueger, Simón Piga, Andrew Treglown
In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha \gt 0$, there exists a constant $C$ such that for every $n$-vertex digraph of minimum semi-degree at least $\alpha n$, if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle
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On the choosability of -minor-free graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-11-03 Olivier Fischer, Raphael Steiner
Given a graph $H$, let us denote by $f_\chi (H)$ and $f_\ell (H)$, respectively, the maximum chromatic number and the maximum list chromatic number of $H$-minor-free graphs. Hadwiger’s famous colouring conjecture from 1943 states that $f_\chi (K_t)=t-1$ for every $t \ge 2$. A closely related problem that has received significant attention in the past concerns $f_\ell (K_t)$, for which it is known that
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Spanning subdivisions in Dirac graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-10-20 Matías Pavez-Signé
We show that for every $n\in \mathbb N$ and $\log n\le d\lt n$, if a graph $G$ has $N=\Theta (dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every $n$-vertex $d$-regular graph.
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Intersecting families without unique shadow Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-10-02 Peter Frankl, Jian Wang
Let $\mathcal{F}$ be an intersecting family. A $(k-1)$-set $E$ is called a unique shadow if it is contained in exactly one member of $\mathcal{F}$. Let ${\mathcal{A}}=\{A\in \binom{[n]}{k}\colon |A\cap \{1,2,3\}|\geq 2\}$. In the present paper, we show that for $n\geq 28k$, $\mathcal{A}$ is the unique family attaining the maximum size among all intersecting families without unique shadow. Several other
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Many Hamiltonian subsets in large graphs with given density Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-10-02 Stijn Cambie, Jun Gao, Hong Liu
A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh, and Staden proved that for large $d$, among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes
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The Excluded Tree Minor Theorem Revisited Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-09-27 Vida Dujmović, Robert Hickingbotham, Gwenaël Joret, Piotr Micek, Pat Morin, David R. Wood
We prove that for every tree $T$ of radius $h$, there is an integer $c$ such that every $T$-minor-free graph is contained in $H\boxtimes K_c$ for some graph $H$ with pathwidth at most $2h-1$. This is a qualitative strengthening of the Excluded Tree Minor Theorem of Robertson and Seymour (GM I). We show that radius is the right parameter to consider in this setting, and $2h-1$ is the best possible bound
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On the zeroes of hypergraph independence polynomials Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-09-21 David Galvin, Gwen McKinley, Will Perkins, Michail Sarantis, Prasad Tetali
We study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is
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Height function localisation on trees Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-09-18 Piet Lammers, Fabio Toninelli
We study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of
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On the size of maximal intersecting families Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-09-08 Dmitrii Zakharov
We show that an $n$-uniform maximal intersecting family has size at most $e^{-n^{0.5+o(1)}}n^n$. This improves a recent bound by Frankl ((2019) Comb. Probab. Comput. 28(5) 733–739.). The Spread Lemma of Alweiss et al. ((2020) Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing.) plays an important role in the proof.
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Forcing generalised quasirandom graphs efficiently Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-09-05 Andrzej Grzesik, Daniel Král’, Oleg Pikhurko
We study generalised quasirandom graphs whose vertex set consists of $q$ parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with
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Quasipolynomial-time algorithms for Gibbs point processes Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-08-11 Matthew Jenssen, Marcus Michelen, Mohan Ravichandran
We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a stable potential. This result holds for all activities $\lambda$ for which the partition function satisfies a zero-free assumption in a neighbourhood of the interval $[0,\lambda ]$. As a corollary, for all finiterange stable potentials, we obtain a quasipolynomial-time
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Limiting empirical spectral distribution for the non-backtracking matrix of an Erdős-Rényi random graph Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-07-31 Ke Wang, Philip Matchett Wood
In this note, we give a precise description of the limiting empirical spectral distribution for the non-backtracking matrices for an Erdős-Rényi graph $G(n,p)$ assuming $np/\log n$ tends to infinity. We show that derandomizing part of the non-backtracking random matrix simplifies the spectrum considerably, and then, we use Tao and Vu’s replacement principle and the Bauer-Fike theorem to show that the
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On the exponential growth rates of lattice animals and interfaces Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-07-31 Agelos Georgakopoulos, Christoforos Panagiotis
We introduce a formula for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice versa. We exploit this in both directions. We obtain the rigorous lower bound ${\dot{p}_c}({\mathbb{Z}}^3)\gt 0.2522$ for 3-dimensional site percolation. We also improve on the best known asymptotic bounds on $a({\mathbb{Z}}^d)$
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On the maximum number of edges in -critical graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-07-24 Cong Luo, Jie Ma, Tianchi Yang
A graph is called $k$-critical if its chromatic number is $k$ but every proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex $k$-critical graph. This is widely open for every integer $k\geq 4$. Using a structural characterisation of Greenwell and Lovász and an extremal result of Simonovits, Stiebitz
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Polarised random -SAT Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-07-20 Joel Larsson Danielsson, Klas Markström
In this paper we study a variation of the random $k$-SAT problem, called polarised random $k$-SAT, which contains both the classical random $k$-SAT model and the random version of monotone $k$-SAT another well-known NP-complete version of SAT. In this model there is a polarisation parameter $p$, and in half of the clauses each variable occurs negated with probability $p$ and pure otherwise, while in
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The codegree Turán density of tight cycles minus one edge Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-07-05 Simón Piga, Marcelo Sales, Bjarne Schülke
Given $\alpha \gt 0$ and an integer $\ell \geq 5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $\alpha n$ edges contains a copy of $C_\ell ^{-}$, a tight cycle on $\ell$ vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidický.
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On the size-Ramsey number of grids Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-06-26 David Conlon, Rajko Nenadov, Miloš Trujić
We show that the size-Ramsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is $O(n^{5/4})$, improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.
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Archaeology of random recursive dags and Cooper-Frieze random networks Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-06-13 Simon Briend, Francisco Calvillo, Gábor Lugosi
We study the problem of finding the root vertex in large growing networks. We prove that it is possible to construct confidence sets of size independent of the number of vertices in the network that contain the root vertex with high probability in various models of random networks. The models include uniform random recursive dags and uniform Cooper-Frieze random graphs.
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On the number of error correcting codes Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-06-09 Dingding Dong, Nitya Mani, Yufei Zhao
We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$, where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3}
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Spanning -cycles in random graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-06-09 Alberto Espuny Díaz, Yury Person
We extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of
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Unimodular random one-ended planar graphs are sofic Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-06-09 Ádám Timár
We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended
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A smoother notion of spread hypergraphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-06-08 Sam Spiro
Alweiss, Lovett, Wu, and Zhang introduced $q$-spread hypergraphs in their breakthrough work regarding the sunflower conjecture, and since then $q$-spread hypergraphs have been used to give short proofs of several outstanding problems in probabilistic combinatorics. A variant of $q$-spread hypergraphs was implicitly used by Kahn, Narayanan, and Park to determine the threshold for when a square of a
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Unavoidable patterns in locally balanced colourings Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-06-01 Nina Kamčev, Alp Müyesser
Which patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours? We show that when $\varepsilon \gt 1/4$, $K_n$ must contain a complete subgraph on $\Omega (\log n)$ vertices where one of the colours forms a balanced complete bipartite graph. When $\varepsilon \leq 1/4$, this statement is no longer true, as evidenced by the following
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Subspace coverings with multiplicities Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-05-18 Anurag Bishnoi, Simona Boyadzhiyska, Shagnik Das, Tamás Mészáros
We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k - 1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$. The value of $f(n,1,1)$ also follows
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A pair degree condition for Hamiltonian cycles in 3-uniform hypergraphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-05-17 Bjarne Schülke
We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$-uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s
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Universal geometric graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-05-15 Fabrizio Frati, Michael Hoffmann, Csaba D. Tóth
We extend the notion of universal graphs to a geometric setting. A geometric graph is universal for a class $\mathcal H$ of planar graphs if it contains an embedding, that is, a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric graph with $n$ vertices and $O\!\left(n \log n\right)$ edges that is universal for $n$-vertex forests; this generalises
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Maximal chordal subgraphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-05-02 Lior Gishboliner, Benny Sudakov
A chordal graph is a graph with no induced cycles of length at least $4$. Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erdős and Laskar posed the problem of estimating $f(n,m)$. In the late 1980s, Erdős, Gyárfás, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture
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Ramsey upper density of infinite graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-04-25 Ander Lamaison
For a fixed infinite graph $H$, we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-colouring of the edges of $K_{\mathbb{N}}$. This is called the Ramsey upper density of $H$ and was introduced by Erdős and Galvin in a restricted setting, and by DeBiasio and McKenney in general. Recently [4], the Ramsey upper density of the infinite path was determined
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Clique-factors in graphs with sublinear -independence number Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-04-24 Jie Han, Ping Hu, Guanghui Wang, Donglei Yang
Given a graph $G$ and an integer $\ell \ge 2$ , we denote by $\alpha _{\ell }(G)$ the maximum size of a $K_{\ell }$ -free subset of vertices in $V(G)$ . A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$ -vertex graphs $G$ with $\alpha _{\ell }(G) = o(n)$ , which can be seen as a Ramsey–Turán variant of the celebrated
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Expected number of faces in a random embedding of any graph is at most linear Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-04-24 Jesse Campion Loth, Bojan Mohar
A random two-cell embedding of a given graph $G$ is obtained by choosing a random local rotation around every vertex. We analyse the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl [5] proved that the expected number of faces in a random embedding of an arbitrary graph of order $n$ is at most $n\log (n)$ . While there are many families
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Problems and results on 1-cross-intersecting set pair systems Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-04-24 Zoltán Füredi, András Gyárfás, Zoltán Király
The notion of cross-intersecting set pair system of size $m$ , $ (\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m )$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne \emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le\binom{a+b}{a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$ . Our central problem is to see how this bound changes
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Abelian groups from random hypergraphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-04-20 Andrew Newman
For a $k$ -uniform hypergraph $\mathcal{H}$ on vertex set $\{1, \ldots, n\}$ we associate a particular signed incidence matrix $M(\mathcal{H})$ over the integers. For $\mathcal{H} \sim \mathcal{H}_k(n, p)$ an Erdős–Rényi random $k$ -uniform hypergraph, ${\mathrm{coker}}(M(\mathcal{H}))$ is then a model for random abelian groups. Motivated by conjectures from the study of random simplicial complexes
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Poset Ramsey numbers: large Boolean lattice versus a fixed poset Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-02-17 Maria Axenovich, Christian Winter
Given partially ordered sets (posets) $(P, \leq _P\!)$ and $(P^{\prime}, \leq _{P^{\prime}}\!)$ , we say that $P^{\prime}$ contains a copy of $P$ if for some injective function $f\,:\, P\rightarrow P^{\prime}$ and for any $X, Y\in P$ , $X\leq _P Y$ if and only if $f(X)\leq _{P^{\prime}} f(Y)$ . For any posets $P$ and $Q$ , the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that
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Multiple random walks on graphs: mixing few to cover many Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-02-15 Nicolás Rivera, Thomas Sauerwald, John Sylvester
Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start
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Random feedback shift registers and the limit distribution for largest cycle lengths Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-02-14 Richard A. Arratia, E. Rodney Canfield, Alfred W. Hales
For a random binary noncoalescing feedback shift register of width $n$ , with all $2^{2^{n-1}}$ possible feedback functions $f$ equally likely, the process of long cycle lengths, scaled by dividing by $N=2^n$ , converges in distribution to the same Poisson–Dirichlet limit as holds for random permutations in $\mathcal{S}_N$ , with all $N!$ possible permutations equally likely. Such behaviour was conjectured
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Bipartite-ness under smooth conditions Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-02-03 Tao Jiang, Sean Longbrake, Jie Ma
Given a family $\mathcal{F}$ of bipartite graphs, the Zarankiewicz number $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such $G$ is called $\mathcal{F}$ -free). For $1\leq \beta \lt \alpha \lt 2$ , a family $\mathcal{F}$ of bipartite graphs is $(\alpha,\beta )$ -smooth if for some $\rho \gt 0$
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Convergence of blanket times for sequences of random walks on critical random graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-01-09 George Andriopoulos
Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$ -blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$ -blanket times of the random walks
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Off-diagonal book Ramsey numbers Comb. Probab. Comput. (IF 0.9) Pub Date : 2023-01-09 David Conlon, Jacob Fox, Yuval Wigderson
The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ . Here we consider the natural off-diagonal variant $r(B_{cn} ^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$ . In this more general setting, we show that an interesting dichotomy emerges: for very small $c$ , a simple
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The bunkbed conjecture holds in the limit Comb. Probab. Comput. (IF 0.9) Pub Date : 2022-12-14 Tom Hutchcroft, Alexander Kent, Petar Nizić-Nikolac
Let $G=(V,E)$ be a countable graph. The Bunkbed graph of $G$ is the product graph $G \times K_2$ , which has vertex set $V\times \{0,1\}$ with “horizontal” edges inherited from $G$ and additional “vertical” edges connecting $(w,0)$ and $(w,1)$ for each $w \in V$ . Kasteleyn’s Bunkbed conjecture states that for each $u,v \in V$ and $p\in [0,1]$ , the vertex $(u,0)$ is at least as likely to be connected
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A bipartite version of the Erdős–McKay conjecture Comb. Probab. Comput. (IF 0.9) Pub Date : 2022-12-09 Eoin Long, Laurenţiu Ploscaru
An old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$ -vertex graph are of order $O(\!\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ . We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\!\log n)$ , then the graph contains induced
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Fluctuations of subgraph counts in graphon based random graphs Comb. Probab. Comput. (IF 0.9) Pub Date : 2022-12-09 Bhaswar B. Bhattacharya, Anirban Chatterjee, Svante Janson
Given a graphon $W$ and a finite simple graph $H$ , with vertex set $V(H)$ , denote by $X_n(H, W)$ the number of copies of $H$ in a $W$ -random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$ . Towards this we introduce a notion
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On random walks and switched random walks on homogeneous spaces Comb. Probab. Comput. (IF 0.9) Pub Date : 2022-11-28 Elvira Moreno, Mauricio Velasco
We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$ . We introduce the switched random walk determined by a finite set of probability distributions on $G$ , prove that its long-term behaviour is determined by the Fourier joint spectral radius of the distributions, and give Hermitian sum-of-squares algorithms for
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Supercritical site percolation on the hypercube: small components are small Comb. Probab. Comput. (IF 0.9) Pub Date : 2022-11-25 Sahar Diskin, Michael Krivelevich
We consider supercritical site percolation on the $d$ -dimensional hypercube $Q^d$ . We show that typically all components in the percolated hypercube, besides the giant, are of size $O(d)$ . This resolves a conjecture of Bollobás, Kohayakawa, and Łuczak from 1994.