arXiv subjects

Michael Krivelevich

Publications and source records attributed to Michael Krivelevich.

At least 145 records · Page 8Linked to original sources

arXiv2013

Long paths and cycles in random subgraphs of graphs with large minimum degree

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge ω/k$ for a function $ω=ω(k)$ that tends to infinity as $k$ does, then $G_p$ asymptotically almost surely contains a cycle (and thus a path) of length at least $(1-o(1))k$, and (ii) if $p \ge (1+o(1))\ln k/k$, then $G_p$ asymptotically almost surely contains a path of length at least $k$. Our theorems extend classical results on paths and cycles in the binomial random graph, obtained by taking $G$ to be the complete graph on $k+1$ vertices.

math.CO↗

arXiv2013

The biased odd cycle game

In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$. We prove that for every $δ>0$ and large enough $n$, there exists a constant $k$ for which if $δ(G)\geq δn$ and $χ(G)\geq k$, then Maker can build an odd cycle in the $(1:b)$ game for $b=O(\frac{n}{\log^2 n})$. We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, Łuczak and Rödl: is it true that for any positive constants $t$ and $b$, there exists an integer $k$ such that for every graph $G$, if $χ(G)\geq k$, then Maker can build a graph which is not $t$-colorable, in the $(1:b)$ Maker-Breaker game played on the vertices of $G$?

math.CO↗

arXiv2013

A construction of almost Steiner systems

Let $n$, $k$, and $t$ be integers satisfying $n>k>t\ge2$. A Steiner system with parameters $t$, $k$, and $n$ is a $k$-uniform hypergraph on $n$ vertices in which every set of $t$ distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for $t\geq6$. In this note we prove that for every $k>t\ge2$ and sufficiently large $n$, there exists an almost Steiner system with parameters $t$, $k$, and $n$; that is, there exists a $k$-uniform hypergraph on $n$ vertices such that every set of $t$ distinct vertices is covered by either one or two edges.

math.CO↗

arXiv2012

Hamilton cycles in random geometric graphs

We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbor model, there is a constant κ such that almost every κ-connected graph has a Hamilton cycle.

math.PR↗

arXiv2012

Biased Games On Random Boards

In this paper we analyze biased Maker-Breaker games and Avoider-Enforcer games, both played on the edge set of a random board $G\sim \gnp$. In Maker-Breaker games there are two players, denoted by Maker and Breaker. In each round, Maker claims one previously unclaimed edge of $G$ and Breaker responds by claiming $b$ previously unclaimed edges. We consider the Hamiltonicity game, the perfect matching game and the $k$-vertex-connectivity game, where Maker's goal is to build a graph which possesses the relevant property. Avoider-Enforcer games are the reverse analogue of Maker-Breaker games with a slight modification, where the two players claim at least 1 and at least $b$ previously unclaimed edges per move, respectively, and Avoider aims to avoid building a graph which possesses the relevant property. Maker-Breaker games are known to be "bias-monotone", that is, if Maker wins the $(1,b)$ game, he also wins the $(1,b-1)$ game. Therefore, it makes sense to define the critical bias of a game, $b^*$, to be the "breaking point" of the game. That is, Maker wins the $(1,b)$ game whenever $b\leq b^*$ and loses otherwise. An analogous definition of the critical bias exists for Avoider-Enforcer games: here, the critical bias of a game $b^*$ is such that Avoider wins the $(1,b)$ game for every $b > b^*$, and loses otherwise. We prove that, for every $p=ω(\frac{\ln n}{n})$, $G\sim\gnp$ is typically such that the critical bias for all the aforementioned Maker-Breaker games is asymptotically $b^*=\frac{np}{\ln n}$. We also prove that in the case $p=Θ(\frac{\ln n}{n})$, the critical bias is $b^*=Θ(\frac{np}{\ln n})$. These results settle a conjecture of Stojaković and Szabó. For Avoider-Enforcer games, we prove that for $p=Ω(\frac{\ln n}{n})$, the critical bias for all the aforementioned games is $b^*=Θ(\frac{np}{\ln n})$.

math.CO↗

arXiv2012

The phase transition in random graphs - a simple proof

The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any ε>0 and p=(1-ε)/n, all connected components of G(n,p) are typically of size O(log n), while for p=(1+ε)/n, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime p=(1+ε)/n, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games.

math.CO↗

arXiv2012

Robust Hamiltonicity of Dirac graphs

A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac's theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability $p$, and prove that there exists a constant $C$ such that if $p \ge C \log n / n$, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a $(1:b)$ Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias $b$ is at most $cn /\log n$ for some absolute constant $c > 0$. Both of these results are tight up to a constant factor, and are proved under one general framework.

math.CO↗

arXiv2012

Fast strategies in Maker-Breaker games played on random boards

In this paper we analyze classical Maker-Breaker games played on the edge set of a sparse random board $G\sim \gnp$. We consider the Hamiltonicity game, the perfect matching game and the $k$-connectivity game. We prove that for $p(n)\geq \text{polylog}(n)/n$, the board $G\sim \gnp$ is typically such that Maker can win these games asymptotically as fast as possible, i.e. within $n+o(n)$, $n/2+o(n)$ and $kn/2+o(n)$ moves respectively.

math.CO↗

arXiv2012

On the number of Hamilton cycles in pseudo-random graphs

We prove that if G is an (n,d,lambda)-graph (a d-regular graph on n vertices, all of whose non-trivial eigenvalues are at most lambda) and the following conditions are satisfied: 1. d/lambda >= (log n)^{1+epsilon} for some constant epsilon>0; 2.log d * lod (d/lambda) >> log n, then the number of Hamilton cycles in G is n!(d/n)^n(1+o(1))^n.

math.CO↗

arXiv2011

On covering expander graphs by Hamilton cycles

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree $Δ$ satisfies some basic expansion properties and contains a family of $(1-o(1))Δ/2$ edge disjoint Hamilton cycles, then there also exists a covering of its edges by $(1+o(1))Δ/2$ Hamilton cycles. This implies that for every $α>0$ and every $p \geq n^{α-1}$ there exists a covering of all edges of $G(n,p)$ by $(1+o(1))np/2$ Hamilton cycles asymptotically almost surely, which is nearly optimal.

math.CO↗

arXiv2011

Optimal packings of Hamilton cycles in sparse random graphs

We prove that there exists a positive constant εsuch that if \log n / n \le p \le n^{-1+ε}, then asymptotically almost surely the random graph G ~ G(n,p) contains a collection of \lfloor δ(G)/2 \rfloor edge-disjoint Hamilton cycles.

math.CO↗

arXiv2011

Expanders Are Universal for the Class of All Spanning Trees

Given a class of graphs F, we say that a graph G is universal for F, or F-universal, if every H in F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of attention. One is particularly interested in tight F-universal graphs, i.e., graphs whose number of vertices is equal to the largest number of vertices in a graph from F. Arguably, the most studied case is that when F is some class of trees. Given integers n and Δ, we denote by T(n,Δ) the class of all n-vertex trees with maximum degree at most Δ. In this work, we show that every n-vertex graph satisfying certain natural expansion properties is T(n,Δ)-universal or, in other words, contains every spanning tree of maximum degree at most Δ. Our methods also apply to the case when Δis some function of n. The result has a few very interesting implications. Most importantly, we obtain that the random graph G(n,p) is asymptotically almost surely (a.a.s.) universal for the class of all bounded degree spanning (i.e., n-vertex) trees provided that p \geq c n^{-1/3} \log^2n where c > 0 is a constant. Moreover, a corresponding result holds for the random regular graph of degree pn. In fact, we show that if Δsatisfies \log n \leq Δ\leq n^{1/3}, then the random graph G(n,p) with p \geq c Δn^{-1/3} \log n and the random r-regular n-vertex graph with r \geq cΔn^{2/3} \log n are a.a.s. T(n,Δ)-universal. Another interesting consequence is the existence of locally sparse n-vertex T(n,Δ)-universal graphs. For constant Δ, we show that one can (randomly) construct n-vertex T(n,Δ)-universal graphs with clique number at most five. Finally, we show robustness of random graphs with respect to being universal for T(n,Δ) in the context of the Maker-Breaker tree-universality game.

math.CO↗

arXiv2011

Biased orientation games

We study biased {\em orientation games}, in which the board is the complete graph $K_n$, and Maker and Breaker take turns in directing previously undirected edges of $K_n$. At the end of the game, the obtained graph is a tournament. Maker wins if the tournament has some property $\mathcal P$ and Breaker wins otherwise. We provide bounds on the bias that is required for a Maker's win and for a Breaker's win in three different games. In the first game Maker wins if the obtained tournament has a cycle. The second game is Hamiltonicity, where Maker wins if the obtained tournament contains a Hamilton cycle. Finally, we consider the $H$-creation game, where Maker wins if the obtained tournament has a copy of some fixed graph $H$.

math.CO↗

arXiv2011

Longest cycles in sparse random digraphs

Long paths and cycles in sparse random graphs and digraphs were studied intensively in the 1980's. It was finally shown by Frieze in 1986 that the random graph $\cG(n,p)$ with $p=c/n$ has a cycle on at all but at most $(1+ε)ce^{-c}n$ vertices with high probability, where $ε=ε(c)\to 0$ as $c\to\infty$. This estimate on the number of uncovered vertices is essentially tight due to vertices of degree 1. However, for the random digraph $\cD(n,p)$ no tight result was known and the best estimate was a factor of $c/2$ away from the corresponding lower bound. In this work we close this gap and show that the random digraph $\cD(n,p)$ with $p=c/n$ has a cycle containing all but $(2+ε)e^{-c}n$ vertices w.h.p., where $ε=ε(c)\to 0$ as $c\to\infty$. This is essentially tight since w.h.p. such a random digraph contains $(2e^{-c}-o(1))n$ vertices with zero in-degree or out-degree.

math.CO↗

arXiv2011

On the resilience of Hamiltonicity and optimal packing of Hamilton cycles in random graphs

Let $\bk=(k_1,...,k_n)$ be a sequence of $n$ integers. For an increasing monotone graph property $\mP$ we say that a base graph $G=([n],E)$ is \emph{$\bk$-resilient} with respect to $\mP$ if for every subgraph $H\subseteq G$ such that $d_H(i)\leq k_i$ for every $1\leq i\leq n$ the graph $G-H$ possesses $\mP$. This notion naturally extends the idea of the \emph{local resilience} of graphs recently initiated by Sudakov and Vu. In this paper we study the $\bk$-resilience of a typical graph from $\GNP$ with respect to the Hamiltonicity property where we let $p$ range over all values for which the base graph is expected to be Hamiltonian. In particular, we prove that for every $ε>0$ and $p\geq\frac{\ln n+\ln\ln n +ω(1)}{n}$ if a graph is sampled from $\GNP$ then with high probability removing from each vertex of "small" degree all incident edges but two and from any other vertex at most a $(\frac{1}{3}-ε)$-fraction of the incident edges will result in a Hamiltonian graph. Considering this generalized approach to the notion of resilience allows to establish several corollaries which improve on the best known bounds of Hamiltonicity related questions. It implies that for every positive $ε>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of $\GNP$ with respect to being Hamiltonian is at least $(1-ε)np/3$, improving on the previous bound for this range of $p$. Another implication is a result on optimal packing of edge disjoint Hamilton cycles in a random graph. We prove that if $p\leq\frac{1.02\ln n}{n}$ then with high probability a graph $G$ sampled from $\GNP$ contains $\lfloor\frac{δ(G)}{2}\rfloor$ edge disjoint Hamilton cycles, extending the previous range of $p$ for which this was known to hold.

math.CO↗

arXiv2010

Fast embedding of spanning trees in biased Maker-Breaker games

Given a tree $T=(V,E)$ on $n$ vertices, we consider the $(1 : q)$ Maker-Breaker tree embedding game ${\mathcal T}_n$. The board of this game is the edge set of the complete graph on $n$ vertices. Maker wins ${\mathcal T}_n$ if and only if he is able to claim all edges of a copy of $T$. We prove that there exist real numbers $α, ε> 0$ such that, for sufficiently large $n$ and for every tree $T$ on $n$ vertices with maximum degree at most $n^ε$, Maker has a winning strategy for the $(1 : q)$ game ${\mathcal T}_n$, for every $q \leq n^α$. Moreover, we prove that Maker can win this game within $n + o(n)$ moves which is clearly asymptotically optimal.

math.CO↗

arXiv2010

Long cycles in subgraphs of (pseudo)random directed graphs

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every $0 < γ< 1/2$ we find a constant $c=c(γ)$ such that the following holds. Let $G=(V,E)$ be a (pseudo)random directed graph on $n$ vertices, and let $G'$ be a subgraph of $G$ with $(1/2+γ)|E|$ edges. Then $G'$ contains a directed cycle of length at least $(c-o(1))n$. Moreover, there is a subgraph $G''$ of $G$ with $(1/2+γ-o(1))|E|$ edges that does not contain a cycle of length at least $cn$.

math.CO↗

arXiv2010

Embedding spanning trees in random graphs

We prove that if T is a tree on n vertices wih maximum degree D and the edge probability p(n) satisfies: np>c*max{D*logn,n^ε} for some constant ε>0, then with high probability the random graph G(n,p) contains a copy of T. The obtained bound on the edge probability is shown to be essentially tight for D=n^{Θ(1)}.

math.CO↗