arXiv subjects

Michael Krivelevich

Publications and source records attributed to Michael Krivelevich.

At least 109 records · Page 6Linked to original sources

arXiv2017

On MAXCUT in strictly supercritical random graphs, and coloring of random graphs and random tournaments

We use a theorem by Ding, Lubetzky and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of $G\sim G\left(n,\frac {1+\varepsilon}n\right)$ in terms of $\varepsilon$. We then apply this result to prove the following conjecture by Frieze and Pegden. For every $\varepsilon>0$ there exists $\ell_\varepsilon$ such that \whp $G\sim G(n,\frac {1+\varepsilon}n)$ is not homomorphic to the cycle on $2\ell_\varepsilon+1$ vertices. We also consider the coloring properties of biased random tournaments. A $p$-random tournament on $n$ vertices is obtained from the transitive tournament by reversing each edge independently with probability $p$. We show that for $p=Θ(\frac 1n)$ the chromatic number of a $p$-random tournament behaves similarly to that of a random graph with the same edge probability. To treat the case $p=\frac {1+\varepsilon}n$ we use the aforementioned result on MAXCUT.

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arXiv2017

Waiter-Client and Client-Waiter Hamiltonicity games on random graphs

We study two types of two player, perfect information games with no chance moves, played on the edge set of the binomial random graph ${\mathcal G}(n,p)$. In each round of the $(1 : q)$ Waiter-Client Hamiltonicity game, the first player, called Waiter, offers the second player, called Client, $q+1$ edges of ${\mathcal G}(n,p)$ which have not been offered previously. Client then chooses one of these edges, which he claims, and the remaining $q$ edges go back to Waiter. Waiter wins this game if by the time every edge of ${\mathcal G}(n,p)$ has been claimed by some player, the graph consisting of Client's edges is Hamiltonian; otherwise Client is the winner. Client-Waiter games are defined analogously, the main difference being that Client wins the game if his graph is Hamiltonian and Waiter wins otherwise. In this paper we determine a sharp threshold for both games. Namely, for every fixed positive integer $q$, we prove that the smallest edge probability $p$ for which a.a.s. Waiter has a winning strategy for the $(1 : q)$ Waiter-Client Hamiltonicity game is $(1 + o(1)) \log n/n$, and the smallest $p$ for which a.a.s. Client has a winning strategy for the $(1 : q)$ Client-Waiter Hamiltonicity game is $(q + 1 + o(1)) \log n/n$.

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arXiv2016

Small subgraphs in the trace of a random walk

We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph $G(n,p)$. In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length $m$ is essentially equal to the threshold for its appearance in the random graph drawn from $G(n,m)$. In the case where the base graph is the complete graph, we show that a fixed forest appears in the trace typically much earlier than it appears in $G(n,m)$.

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arXiv2016

Bounded-degree spanning trees in randomly perturbed graphs

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizeable body of existing research on randomly perturbed graphs. Specifically, we show that there is $c = c(α,Δ)$ such that if G is an n-vertex graph with minimum degree at least $αn$, and T is an n-vertex tree with maximum degree at most $Δ$ , then if we add cn uniformly random edges to G, the resulting graph will contain T asymptotically almost surely (as $n\to\infty$ ). Our proof uses a lemma concerning the decomposition of a dense graph into super-regular pairs of comparable sizes, which may be of independent interest.

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arXiv2016

Finding Hamilton cycles in random graphs with few queries

We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of ${\mathcal G}(n,p)$ in order to typically find a subgraph possessing a given target property. We show that if $p\geq \frac{\ln n+\ln\ln n+ω(1)}{n}$, then one can find a Hamilton cycle with high probability after exposing $(1+o(1))n$ edges. Our result is tight in both $p$ and the number of exposed edges.

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arXiv2016

Finding paths in sparse random graphs requires many queries

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in $G\sim \mathcal G(n,p)$ when $p=\frac{1+\varepsilon}{n}$ for some fixed constant $\varepsilon>0$. This random graph is known to have typically linearly long paths. To have $\ell$ edges with high probability in $G\sim \mathcal G(n,p)$ one clearly needs to query at least $Ω\left(\frac{\ell}{p}\right)$ pairs of vertices. Can we find a path of length $\ell$ economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length $\ell=Ω\left(\frac{\log\left(\frac{1}{\varepsilon}\right)}{\varepsilon}\right)$ with at least constant probability in $G\sim \mathcal G(n,p)$ with $p=\frac{1+\varepsilon}{n}$ must query at least $Ω\left(\frac{\ell}{p\varepsilon \log\left(\frac{1}{\varepsilon}\right)}\right)$ pairs of vertices. This is tight up to the $\log\left(\frac{1}{\varepsilon}\right)$ factor.

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arXiv2016

The emergence of a giant component in random subgraphs of pseudo-random graphs

Let $G$ be a $d$-regular graph $G$ on $n$ vertices. Suppose that the adjacency matrix of $G$ is such that the eigenvalue $λ$ which is second largest in absolute value satisfies $λ=o(d)$. Let $G_p$ with $p=\fracα{d}$ be obtained from $G$ by including each edge of $G$ independently with probability $p$. We show that if $α<1$ then whp the maximum component size of $G_p$ is $O(\log n)$ and if $α>1$ then $G_p$ contains a unique giant component of size $Ω(n)$, with all other components of size $O(\log n)$.

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arXiv2016

Adding random edges to dense graphs

This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small diameter and $k$-connectivity.

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arXiv2016

Client-Waiter games on complete and random graphs

For a graph G, a monotone increasing graph property P and positive integer q, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most q+1 unclaimed edges of G from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property P by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the H-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the H-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.

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arXiv2016

Random-Player Maker-Breaker games

In a $(1:b)$ Maker-Breaker game, a primary question is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erdős conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, "Erdős Paradigm" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly. A random-player Maker-Breaker game is a two-player game, played the same as an ordinary (biased) Maker-Breaker game, except that one player plays according to his best strategy and claims one element in each round, while the other plays randomly and claims $b$ elements. In fact, for every (ordinary) Maker-Breaker game, there are two different random-player versions; the $(1:b)$ random-Breaker game and the $(m:1)$ random-Maker game. We analyze the random-player version of several classical Maker-Breaker games such as the Hamilton cycle game, the perfect-matching game and the $k$-vertex-connectivity game (played on the edge sets of $K_n$). For each of these games we find or estimate the asymptotic values of $b$ that allow each player to typically win the game. In fact, we provide the "smart" player with an explicit winning strategy for the corresponding value of $b$.

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arXiv2016

Cycles and matchings in randomly perturbed digraphs and hypergraphs

We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

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arXiv2016

Contagious Sets in Random Graphs

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors. A \emph{contagious set} is a set whose activation results with the entire graph being active. Given a graph $G$, let $m(G,r)$ be the minimal size of a contagious set. We study this process on the binomial random graph $G:=G(n,p)$ with $p: = \frac{d}{n}$ and $1 \ll d \ll \left(\frac{n \log \log n}{\log^2 n}\right)^{\frac{r-1}{r}}$. Assuming $r > 1$ to be a constant that does not depend on $n$, we prove that $$m(G,r) = Θ\left(\frac{n}{d^{\frac{r}{r-1}}\log d}\right),$$ with high probability. We also show that the threshold probability for $m(G,r)=r$ to hold is $p^*=Θ\left(\frac{1}{(n \log^{r-1} n)^{1/r}}\right)$.

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arXiv2015

Counting and packing Hamilton cycles in dense graphs and oriented graphs

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly $cn$-regular oriented graph on $n$ vertices with $c>3/8$ contains $(cn/e)^n(1+o(1))^n$ directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph $G$ on $n$ vertices of minimum degree at least $(1/2+\varepsilon)n$ contains at least $(1-\varepsilon)\textrm{reg}_{even}(G)/2$ edge-disjoint Hamilton cycles, where $\reg(G)$ is the maximum \emph{even} degree of a spanning regular subgraph of $G$. This establishes an approximate version of a conjecture of Kühn, Lapinskas and Osthus.

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arXiv2015

Manipulative waiters with probabilistic intuition

For positive integers $n$ and $q$ and a monotone graph property $\cA$, we consider the two player, perfect information game $\WC(n,q,\cA)$, which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client, $q+1$ edges of the complete graph $K_n$ which have not been offered previously. Client then chooses one of these edges which he keeps and the remaining $q$ edges go back to Waiter. If at the end of the game, the graph which consists of the edges chosen by Client satisfies the property $\cA$, then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker-Chooser games) for a variety of natural graph theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameter $q$ is close to $n$ and is reminiscent of phase transition phenomena in random graphs. Namely, we prove that if $q \leq (1 - \varepsilon) n$, then Client can avoid connected components of order $c \varepsilon^{-2} \ln n$ for some absolute constant $c > 0$, whereas, for $q \geq (1 + \varepsilon) n$, Waiter can force a giant, linearly sized, connected component in Client's graph. We also prove that Waiter can force Client's graph to be pancyclic for every $q \leq c n$, where $c > 0$ is an appropriate constant.

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arXiv2015

Waiter-Client and Client-Waiter planarity, colorability and minor games

For a finite set $X$, a family of sets ${\mathcal F} \subseteq 2^X$ and a positive integer $q$, we consider two types of two player, perfect information games with no chance moves. In each round of the $(1 : q)$ Waiter-Client game $(X, {\mathcal F})$, the first player, called Waiter, offers the second player, called Client, $q+1$ elements of the board $X$ which have not been offered previously. Client then chooses one of these elements which he claims and the remaining $q$ elements to go back to Waiter. Waiter wins this game if by the time every element of $X$ has been claimed by some player, Client has claimed all elements of some $A \in {\mathcal F}$; otherwise Client is the winner. Client-Waiter games are defined analogously, the main difference being that Client wins the game if he manages to claim all elements of some $A \in {\mathcal F}$ and Waiter wins otherwise. In this paper we study the Waiter-Client and Client-Waiter versions of the non-planarity, $K_t$-minor and non-$k$-colorability games. For each such game, we give a fairly precise estimate of the unique integer $q$ at which the outcome of the game changes from Client's win to Waiter's win. We also discuss the relation between our results, random graphs, and the corresponding Maker-Breaker and Avoider-Enforcer games.

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arXiv2015

Compatible Hamilton cycles in random graphs

A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability $p \gg \frac{\log n}{n}$, the random graph $G(n,p)$ is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph $G=(V,E)$, an {\em incompatibility system} $\mathcal{F}$ over $G$ is a family $\mathcal{F}=\{F_v\}_{v\in V}$ where for every $v\in V$, the set $F_v$ is a set of unordered pairs $F_v \subseteq \{\{e,e'\}: e\ne e'\in E, e\cap e'=\{v\}\}$. An incompatibility system is {\em $Δ$-bounded} if for every vertex $v$ and an edge $e$ incident to $v$, there are at most $Δ$ pairs in $F_v$ containing $e$. We say that a cycle $C$ in $G$ is {\em compatible} with $\mathcal{F}$ if every pair of incident edges $e,e'$ of $C$ satisfies $\{e,e'\} \notin F_v$. This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be used as a quantitative measure of robustness of graph properties. We prove that there is a constant $μ>0$ such that the random graph $G=G(n,p)$ with $p(n) \gg \frac{\log n}{n}$ is asymptotically almost surely such that for any $μnp$-bounded incompatibility system $\mathcal{F}$ over $G$, there is a Hamilton cycle in $G$ compatible with $\mathcal{F}$. We also prove that for larger edge probabilities $p(n)\gg \frac{\log^8n}{n}$, the parameter $μ$ can be taken to be any constant smaller than $1-\frac{1}{\sqrt 2}$. These results imply in particular that typically in $G(n,p)$ for $p \gg \frac{\log n}{n}$, for any edge-coloring in which each color appears at most $μnp$ times at each vertex, there exists a properly colored Hamilton cycle.

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arXiv2015

Some Remarks on Rainbow Connectivity

An edge (vertex) coloured graph is rainbow-connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colours. Rainbow edge (vertex) connectivity of a graph $G$ is the smallest number of colours needed for a rainbow edge (vertex) colouring of $G$. In this paper we propose a very simple approach to studying rainbow connectivity in graphs. Using this idea, we give a unified proof of several known results, as well as some new ones.

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arXiv2015

Generating random graphs in biased Maker-Breaker games

We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for $b=o\left(\sqrt{n}\right)$, Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a $(1:b)$ game on $E(K_n)$. As another application, we show that for $b=Θ\left(n/\ln n\right)$, playing a $(1:b)$ game on $E(K_n)$, Maker can build a graph which contains copies of all spanning trees having maximum degree $Δ=O(1)$ with a bare path of linear length (a bare path in a tree $T$ is a path with all interior vertices of degree exactly two in $T$).

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