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Michael Krivelevich

Publications and source records attributed to Michael Krivelevich.

At least 37 records · Page 2Linked to original sources
arXiv2025

Colouring random subgraphs

We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many $k$-regular graphs $G$ for which the colouring number (i.e., degeneracy) of $G_{1/2}$ is at most $k/3 + o(k)$ with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least $k/2 - o(k)$.

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arXiv2024

Minimum degree conditions for graph rigidity

We study minimum degree conditions that guarantee that an $n$-vertex graph is rigid in $\mathbb{R}^d$. For small values of $d$, we obtain a tight bound: for $d = O(\sqrt{n})$, every $n$-vertex graph with minimum degree at least $(n+d)/2 - 1$ is rigid in $\mathbb{R}^d$. For larger values of $d$, we achieve an approximate result: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $(n+2d)/2 - 1$ is rigid in $\mathbb{R}^d$. This bound is tight up to a factor of two in the coefficient of $d$. As a byproduct of our proof, we also obtain the following result, which may be of independent interest: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $d$ has pseudoachromatic number at least $d+1$; namely, the vertex set of such a graph can be partitioned into $d+1$ subsets such that there is at least one edge between each pair of subsets. This is tight.

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arXiv2024

Fast construction on a restricted budget

We introduce a model of a controlled random graph process. In this model, the edges of the complete graph $K_n$ are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by parameter $t$, and the total budget of purchased edges is bounded by parameter $b$. Builder's goal is to devise a strategy that, with high probability, allows him to construct a graph of purchased edges possessing a target graph property $\mathcal{P}$, all within the limitations of observation time and total budget. We show the following: (a) Builder has a strategy to achieve $k$-vertex-connectivity at the hitting time for this property by purchasing at most $c_kn$ edges for an explicit $c_k 1$; this is optimal in the sense that $C$ cannot be arbitrarily close to $1$. This substantially extends the classical hitting time result for Hamiltonicity due to Ajtai--Komlós--Szemerédi and Bollobás. (c) Builder has a strategy to create a perfect matching by time $(1+\varepsilon)n\log{n}/2$ while purchasing at most $(1+\varepsilon)n/2$ edges (which is optimal). (d) Builder has a strategy to create a copy of a given $k$-vertex tree if $t\ge b\gg\max\{(n/t)^{k-2},1\}$, and this is optimal; (e) For $\ell=2k+1$ or $\ell=2k+2$, Builder has a strategy to create a copy of a cycle of length $\ell$ if $b\gg\max \{n^{k+2}/t^{k+1},n/\sqrt{t}\}$, and this is optimal.

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arXiv2024

Disjoint connected dominating sets in pseudorandom graphs

A connected dominating set (CDS) in a graph is a dominating set of vertices that induces a connected subgraph. Having many disjoint CDSs in a graph can be considered as a measure of its connectivity, and has various graph-theoretic and algorithmic implications. We show that $d$-regular (weakly) pseudoreandom graphs contain $(1+o(1))d/\ln d$ disjoint CDSs, which is asymptotically best possible. In particular, this implies that random $d$-regular graphs typically contain $(1+o(1))d/\ln d$ disjoint CDSs.

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arXiv2024

Components, large and small, are as they should be II: supercritical percolation on regular graphs of constant degree

Let $d\ge 3$ be a fixed integer. Let $y:= y(p)$ be the probability that the root of an infinite $d$-regular tree belongs to an infinite cluster after $p$-bond-percolation. We show that for every constants $b,α>0$ and $1<λ< d-1$, there exist constants $c,C>0$ such that the following holds. Let $G$ be a $d$-regular graph on $n$ vertices, satisfying that for every $U\subseteq V(G)$ with $|U|\le \frac{n}{2}$, $e(U,U^c)\ge b|U|$ and for every $U\subseteq V(G)$ with $|U|\le \log^Cn$, $e(U)\le (1+c)|U|$. Let $p=\fracλ{d-1}$. Then, with probability tending to one as $n$ tends to infinity, the largest component $L_1$ in the random subgraph $G_p$ of $G$ satisfies $\left|1-\frac{|L_1|}{yn}\right|\le α$, and all the other components in $G_p$ are of order $O\left(\frac{λ\log n}{(λ-1)^2}\right)$. This generalises (and improves upon) results for random $d$-regular graphs.

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arXiv2024

Reconstructing random graphs from distance queries

We estimate the minimum number of distance queries that is sufficient to reconstruct the binomial random graph $G(n,p)$ with constant diameter with high probability. We get a tight (up to a constant factor) answer for all $p>n^{-1+o(1)}$ outside "threshold windows" around $n^{-k/(k+1)+o(1)}$, $k\in\mathbb{Z}_{>0}$: with high probability the query complexity equals $Θ(n^{4-d}p^{2-d})$, where $d$ is the diameter of the random graph. This demonstrates the following non-monotone behaviour: the query complexity jumps down at moments when the diameter gets larger; yet, between these moments the query complexity grows. We also show that there exists a non-adaptive algorithm that reconstructs the random graph with $O(n^{4-d}p^{2-d}\ln n)$ distance queries with high probability, and this is best possible.

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arXiv2024

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

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

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arXiv2024

Long cycles in percolated expanders

Given a graph $G$ and probability $p$, we form the random subgraph $G_p$ by retaining each edge of $G$ independently with probability $p$. Given $d\in\mathbb{N}$ and constants $0 0$, we show that if every subset $S\subseteq V(G)$ of size exactly $\frac{c|V(G)|}{d}$ satisfies $|N(S)|\ge d|S|$ and $p=\frac{1+\varepsilon}{d}$, then the probability that $G_p$ does not contain a cycle of length $Ω(\varepsilon^2c^2|V(G)|)$ is exponentially small in $|V(G)|$. As an intermediate step, we also show that given $k,d\in \mathbb{N}$ and a constant $\varepsilon>0$, if every subset $S\subseteq V(G)$ of size exactly $k$ satisfies $|N(S)|\ge kd$ and $p=\frac{1+\varepsilon}{d}$, then the probability that $G_p$ does not contain a path of length $Ω(\varepsilon^2 kd)$ is exponentially small. We further discuss applications of these results to $K_{s,t}$-free graphs of maximal density.

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arXiv2024

The power of many colours

A classical problem, due to Gerencsér and Gyárfás from 1967, asks how large a monochromatic connected component can we guarantee in any $r$-edge colouring of $K_n$? We consider how big a connected component can we guarantee in any $r$-edge colouring of $K_n$ if we allow ourselves to use up to $s$ colours. This is actually an instance of a more general question of Bollobás from about 20 years ago which asks for a $k$-connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided $n$ is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form. We also consider a generalisation in a similar direction of a question first considered by Erdős and Rényi in 1956, who considered given $n$ and $m$, what is the smallest number of $m$-cliques which can cover all edges of $K_n$? This problem is essentially equivalent to the question of what is the minimum number of vertices that are certain to be incident to at least one edge of some colour in any $r$-edge colouring of $K_n$. We consider what happens if we allow ourselves to use up to $s$ colours. We obtain a more complete understanding of the answer to this question for large $n$, in particular determining it up to a constant factor for all $1\le s \le r$, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.

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arXiv2024

Hitting time of connectedness in the random hypercube process

We present a short and self-contained proof of a classical result due to Bollobás (1990): in the random hypercube process, with high probability the hitting time of connectedness equals the hitting time of having minimum degree at least one.

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arXiv2024

Percolation on High-dimensional Product Graphs

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

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arXiv2024

Colouring graphs from random lists

Given positive integers $k \leq m$ and a graph $G$, a family of lists $L = \{L(v) : v \in V(G)\}$ is said to be a random $(k,m)$-list-assignment if for every $v \in V(G)$ the list $L(v)$ is a subset of $\{1, \ldots, m\}$ of size $k$, chosen uniformly at random and independently of the choices of all other vertices. An $n$-vertex graph $G$ is said to be a.a.s. $(k,m)$-colourable if $\lim_{n \to \infty} \mathbb{P}(G \textrm{ is } L-colourable) = 1$, where $L$ is a random $(k,m)$-list-assignment. We prove that if $m \gg n^{1/k^2} Δ^{1/k}$ and $m \geq 3 k^2 Δ$, where $Δ$ is the maximum degree of $G$ and $k \geq 3$ is an integer, then $G$ is a.a.s. $(k,m)$-colourable. This is not far from being best possible, forms a continuation of the so-called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Additionally, we consider this problem under the additional assumption that $G$ is $H$-free for some graph $H$. For various graphs $H$, we estimate the smallest $m$ for which an $H$-free $n$-vertex graph $G$ is a.a.s. $(k,m)$-colourable. This extends and improves several results of Casselgren.

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arXiv2024

Percolation on Irregular High-dimensional Product Graphs

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

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arXiv2024

Isoperimetric Inequalities and Supercritical Percolation on High-dimensional Graphs

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

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arXiv2024

Percolation through Isoperimetry

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

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arXiv2023

Hamilton completion and the path cover number of sparse random graphs

We prove that for every $\varepsilon > 0$ there is $c_0$ such that if $G\sim G(n,c/n)$, $c\ge c_0$, then with high probability $G$ can be covered by at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ vertex disjoint paths, which is essentially tight. This is equivalent to showing that, with high probability, at most $(1+\varepsilon)\cdot \frac{1}{2}ce^{-c} \cdot n$ edges can be added to $G$ to create a Hamiltonian graph.

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arXiv2023

Rigid partitions: from high connectivity to random graphs

A graph is called $d$-rigid if there exists a generic embedding of its vertex set into $\mathbb{R}^d$ such that every continuous motion of the vertices that preserves the lengths of all edges actually preserves the distances between all pairs of vertices. The rigidity of a graph is the maximal $d$ such that the graph is $d$-rigid. We present new sufficient conditions for the $d$-rigidity of a graph in terms of the existence of ``rigid partitions'' -- partitions of the graph that satisfy certain connectivity properties. This extends previous results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz. As an application, we present new results on the rigidity of highly-connected graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense graphs. In particular, we prove that random $C d\log d$-regular graphs are typically $d$-rigid, demonstrate the existence of a giant $d$-rigid component in sparse random binomial graphs, and show that the rigidity of relatively sparse random binomial bipartite graphs is roughly the same as that of the complete bipartite graph, which we consider an interesting phenomenon. Furthermore, we show that a graph admitting $\binom{d+1}{2}$ disjoint connected dominating sets is $d$-rigid. This implies a weak version of the Lovász--Yemini conjecture on the rigidity of highly-connected graphs. We also present an alternative short proof for a recent result by Lew, Nevo, Peled, and Raz, which asserts that the hitting time for $d$-rigidity in the random graph process typically coincides with the hitting time for minimum degree $d$.

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