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Michael Krivelevich

Publications and source records attributed to Michael Krivelevich.

At least 91 records · Page 5Linked to original sources
arXiv2020

Asymptotics in percolation on high-girth expanders

We consider supercritical bond percolation on a family of high-girth $d$-regular expanders. Alon, Benjamini and Stacey (2004) established that its critical probability for the appearance of a linear-sized ("giant'') component is $p_c=1/(d-1)$. Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant and its 2-core at any $p>p_c$. It was further shown in [ABS04] that the second largest component, at any $0<p<1$, has size at most $n^ω$ for some $ω<1$. We show that, unlike the situation in the classical Erdős-Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size $n^{ω'}$ for $ω'$ arbitrarily close to $1$. Moreover, as a by-product of that construction, we answer negatively a question of Benjamini (2013) on the relation between the diameter of a component in percolation on expanders and the existence of a giant component. Finally, we establish other typical features of the giant component, e.g., the existence of a linear path.

math.PR↗
arXiv2019

Optimal shattering of complex networks

We consider optimal attacks or immunization schemes on different models of random graphs. We derive bounds for the minimum number of nodes needed to be removed from a network such that all remaining components are fragments of negligible size. We obtain bounds for different regimes of random regular graphs, Erdős-Rényi random graphs, and scale free networks, some of which are tight. We show that the performance of attacks by degree is bounded away from optimality. Finally we present a polynomial time attack algorithm and prove its optimal performance in certain cases.

physics.soc-ph↗
arXiv2019

Edge-statistics on large graphs

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$ vertices can have. Clearly, this number is $\binom{n}{k}$ for every $n$, $k$ and $\ell \in \left \{0, \binom{k}{2} \right\}$. We conjecture that for every $n$, $k$ and $0 < \ell < \binom{k}{2}$ this number is at most $\left(1/e + o_k(1) \right) \binom{n}{k}$. If true, this would be tight for $\ell \in \{1, k-1\}$. In support of our `Edge-statistics conjecture' we prove that the corresponding density is bounded away from $1$ by an absolute constant. Furthermore, for various ranges of the values of $\ell$ we establish stronger bounds. In particular, we prove that for `almost all' pairs $(k, \ell)$ only a polynomially small fraction of the $k$-subsets of $V(G)$ has exactly $\ell$ edges, and prove an upper bound of $(1/2 + o_k(1))\binom{n}{k}$ for $\ell = 1$. Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun's sieve, as well as graph-theoretic and combinatorial arguments such as Zykov's symmetrization, Sperner's theorem and various counting techniques.

math.CO↗
arXiv2019

Finding a Hamilton cycle fast on average using rotations and extensions

We present an algorithm CRE, which either finds a Hamilton cycle in a graph $G$ or determines that there is no such cycle in the graph. The algorithm's expected running time over input distribution $G\sim G(n,p)$ is $(1+o(1))n/p$, the optimal possible expected time, for $p=p(n) \geq 70n^{-\frac{1}{2}}$. This improves upon previous results on this problem due to Gurevich and Shelah, and to Thomason.

math.CO↗
arXiv2019

Complete minors in graphs without sparse cuts

We show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d = d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order \[ Ω\left( \sqrt{\frac{n d}{\log d}} \right). \] As a corollary we determine the order of a largest complete minor one can guarantee in $d$-regular graphs for which the second largest eigenvalue is bounded away from $d/2$, in $(d/n, o(d))$-jumbled graphs, and in random $d$-regular graphs, for almost all $d = d(n)$.

math.CO↗
arXiv2019

The genus of the Erdős-Rényi random graph and the fragile genus property

We investigate the genus $g(n,m)$ of the Erdős-Rényi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which `region' $m$ falls into. Results already exist for $m \le \frac{n}{2} + O(n^{2/3})$ and $m = ω\left( n^{1+\frac{1}{j}} \right)$ for $j \in \mathbb{N}$, and so we focus on the intermediate cases. We establish that $g(n,m) = (1+o(1)) \frac{m}{2}$ whp (with high probability) when $n \ll m = n^{1+o(1)}$, that $g(n,m) = (1+o(1)) μ(λ) m$ whp for a given function $μ(λ)$ when $m \sim λn$ for $λ> \frac{1}{2}$, and that $g(n,m) = (1+o(1)) \frac{8s^{3}}{3n^{2}}$ whp when $m = \frac{n}{2} + s$ for $n^{2/3} \ll s \ll n$. We then also show that the genus of a fixed graph can increase dramatically if a small number of random edges are added. Given any connected graph with bounded maximum degree, we find that the addition of $εn$ edges will whp result in a graph with genus $Ω(n)$, even when $ε$ is an arbitrarily small constant! We thus call this the `fragile genus' property.

math.CO↗
arXiv2019

Goldberg's Conjecture is true for random multigraphs

In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph $G$, the chromatic index $χ'(G)$ satisfies $χ'(G)\leq \max \{Δ(G)+1, \lceilρ(G)\rceil\}$, where $ρ(G)=\max \{\frac {e(G[S])}{\lfloor |S|/2\rfloor} \mid S\subseteq V \}$. We show that their conjecture (in a stronger form) is true for random multigraphs. Let $M(n,m)$ be the probability space consisting of all loopless multigraphs with $n$ vertices and $m$ edges, in which $m$ pairs from $[n]$ are chosen independently at random with repetitions. Our result states that, for a given $m:=m(n)$, $M\sim M(n,m)$ typically satisfies $χ'(G)=\max\{Δ(G),\lceilρ(G)\rceil\}$. In particular, we show that if $n$ is even and $m:=m(n)$, then $χ'(M)=Δ(M)$ for a typical $M\sim M(n,m)$. Furthermore, for a fixed $\varepsilon>0$, if $n$ is odd, then a typical $M\sim M(n,m)$ has $χ'(M)=Δ(M)$ for $m\leq (1-\varepsilon)n^3\log n$, and $χ'(M)=\lceilρ(M)\rceil$ for $m\geq (1+\varepsilon)n^3\log n$.

math.CO↗
arXiv2019

Expanders - how to find them, and what to find in them

A graph $G=(V,E)$ is called an expander if every vertex subset $U$ of size up to $|V|/2$ has an external neighborhood whose size is comparable to $|U|$. Expanders have been a subject of intensive research for more than three decades and have become one of the central notions of modern graph theory. We first discuss the above definition of an expander and its alternatives. Then we present examples of families of expanding graphs and state basic properties of expanders. Next, we introduce a way to argue that a given graph contains a large expanding subgraph. Finally we research properties of expanding graphs, such as existence of small separators, of cycles (including cycle lengths), and embedding of large minors.

math.CO↗
arXiv2018

Waiter-Client Maximum Degree Game

For integers $n, D, q$ we define a two player perfect information game with no chance moves called the Waiter-Client Maximum Degree game. In this game, two players (Waiter and Client) play on the edges of $K_n$ as follows: in each round, Waiter offers $q+1$ edges which have not been previously offered. Client then claims one of these edges, and Waiter claims the rest. When less than $q+1$ edges which have not been offered remain, Waiter claims them all and the game ends. After the game ends, Client wins if in the graph of his edges, there is no vertex with degree at least $D$, and Waiter wins otherwise. For various values of $q = q(n)$, we study the maximum degree of Client's graph obtained by perfect play. We determine the asymptotic value of Client's maximum degree for the cases $q = o\left( \frac{n}{\ln n} \right)$ and $q = ω\left( \frac{n}{\ln n} \right)$. For the unbiased case $q=1$, we prove that when both players play perfectly the maximum degree $D$ Client achieves satisfies: $D = \frac{n}{2} + Θ(\sqrt{n \ln n})$.

math.CO↗
arXiv2018

The random k-matching-free process

Let $\mathcal{P}$ be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty $n$-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that $\mathcal{P}$ is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the $k$-matching-free process, where $\mathcal{P}$ is the property of not containing a matching of size $k$. We are able to analyse the behaviour of this process for a wide range of values of $k$; in particular we prove that if $k=o(n)$ or if $n-2k=o(\sqrt{n}/\log n)$ then this process is likely to terminate in a $k$-matching-free graph with the maximum possible number of edges, as characterised by Erdős and Gallai. We also show that these bounds on $k$ are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime.

math.CO↗
arXiv2018

Packing Hamilton Cycles Online

It is known that w.h.p. the hitting time $τ_{2σ}$ for the random graph process to have minimum degree $2σ$ coincides with the hitting time for $σ$ edge disjoint Hamilton cycles. In this paper we prove an online version of this property. We show that, for a fixed integer $σ\geq 2$, if random edges of $K_n$ are presented one by one then w.h.p. it is possible to color the edges online with $σ$ colors so that at time $τ_{2σ}$, each color class is Hamiltonian.

math.CO↗
arXiv2017

Proper colouring Painter-Builder game

We consider the following two-player game, parametrised by positive integers $n$ and $k$. The game is played between Painter and Builder, alternately taking turns, with Painter moving first. The game starts with the empty graph on $n$ vertices. In each round Painter colours a vertex of her choice by one of the $k$ colours and Builder claims an edge between two previously unconnected vertices. Both players should maintain that during the game the graph admits a proper $k$-colouring. The game ends if either all $n$ vertices have been coloured, or Painter has no legal move. In the former case, Painter wins the game, in the latter one Builder is the winner. We prove that the minimal number of colours $k=k(n)$ allowing Painter's win is of logarithmic order in the number of vertices $n$. Biased versions of the game are also considered.

math.CO↗
arXiv2017

Clique coloring of dense random graphs

The clique chromatic number of a graph G=(V,E) is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random graph G=G(n,1/2) is, with high probability, Ω(log n). This settles a problem of McDiarmid, Mitsche and Pralat who proved that it is O(log n) with high probability.

math.CO↗
arXiv2017

On the trace of random walks on random graphs

We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$ on the random graph $G\sim G(n,p)$ for $p>C\ln{n}/n$ is, with high probability, Hamiltonian and $Θ(\ln{n})$-connected. In the special case $p=1$ (i.e. when $G=K_n$), we show a hitting time result according to which, with high probability, exactly one step after the last vertex has been visited, the trace becomes Hamiltonian, and one step after the last vertex has been visited for the $k$'th time, the trace becomes $2k$-connected.

math.CO↗
arXiv2017

Elegantly colored paths and cycles in edge colored random graphs

We first consider the following problem. We are given a fixed perfect matching $M$ of $[n]$ and we add random edges one at a time until there is a Hamilton cycle containing $M$. We show that w.h.p. the hitting time for this event is the same as that for the first time there are no isolated vertices in the graph induced by the random edges. We then use this result for the following problem. We generate random edges and randomly color them black or white. A path/cycle is said to \emph{zebraic} if the colors alternate along the path. We show that w.h.p. the hitting time for a zebraic Hamilton cycle coincides with every vertex meeting at least one edge of each color. We then consider some related problems and extend to multiple colors. We also briefly consider directed versions.

math.CO↗
arXiv2017

Finding and using expanders in locally sparse graphs

We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants $c_1>c_2>1$, $0<α<1$, a graph $G$ on $n$ vertices is called a $(c_1,c_2,α)$-graph if it has at least $c_1n$ edges, but every vertex subset $W\subset V(G)$ of size $|W|\le αn$ spans less than $c_2|W|$ edges. We prove that every $(c_1,c_2,α)$-graph with bounded degrees contains an induced expander on linearly many vertices. The proof can be made algorithmic. We then discuss several applications of our main result to random graphs, to problems about embedding graph minors, and to positional games.

math.CO↗