arXiv subjects

Michael Krivelevich

Publications and source records attributed to Michael Krivelevich.

At least 127 records · Page 7Linked to original sources

arXiv2015

Smoothed analysis on connected graphs

The main paradigm of smoothed analysis on graphs suggests that for any large graph $G$ in a certain class of graphs, perturbing slightly the edges of $G$ at random (usually adding few random edges to $G$) typically results in a graph having much "nicer" properties. In this work we study smoothed analysis on trees or, equivalently, on connected graphs. Given an $n$-vertex connected graph $G$, form a random supergraph $G^*$ of $G$ by turning every pair of vertices of $G$ into an edge with probability $\fracε{n}$, where $ε$ is a small positive constant. This perturbation model has been studied previously in several contexts, including smoothed analysis, small world networks, and combinatorics. Connected graphs can be bad expanders, can have very large diameter, and possibly contain no long paths. In contrast, we show that if $G$ is an $n$-vertex connected graph then typically $G^*$ has edge expansion $Ω(\frac{1}{\log n})$, diameter $O(\log n)$, vertex expansion $Ω(\frac{1}{\log n})$, and contains a path of length $Ω(n)$, where for the last two properties we additionally assume that $G$ has bounded maximum degree. Moreover, we show that if $G$ has bounded degeneracy, then typically the mixing time of the lazy random walk on $G^*$ is $O(\log^2 n)$. All these results are asymptotically tight.

math.CO↗

arXiv2015

The phase transition in site percolation on pseudo-random graphs

We establish the existence of the phase transition in site percolation on pseudo-random $d$-regular graphs. Let $G=(V,E)$ be an $(n,d,λ)$-graph, that is, a $d$-regular graph on $n$ vertices in which all eigenvalues of the adjacency matrix, but the first one, are at most $λ$ in their absolute values. Form a random subset $R$ of $V$ by putting every vertex $v\in V$ into $R$ independently with probability $p$. Then for any small enough constant $ε>0$, if $p=\frac{1-ε}{d}$, then with high probability all connected components of the subgraph of $G$ induced by $R$ are of size at most logarithmic in $n$, while for $p=\frac{1+ε}{d}$, if the eigenvalue ratio $λ/d$ is small enough as a function of $ε$, then typically $R$ spans a connected component of size at least $\frac{εn}{d}$ and a path of length proportional to $\frac{ε^2n}{d}$.

math.CO↗

arXiv2015

Long paths and Hamiltonicity in random graphs

We discuss several classical results about long paths and Hamilton cycles in random graphs and present accessible versions of their proofs, relying on the Depth First Search (DFS) algorithm and the notion of boosters.

math.CO↗

arXiv2015

Rainbow Hamilton cycles in random graphs and hypergraphs

Let $H$ be an edge colored hypergraph. We say that $H$ contains a \emph{rainbow} copy of a hypergraph $S$ if it contains an isomorphic copy of $S$ with all edges of distinct colors. We consider the following setting. A randomly edge colored random hypergraph $H\sim \mathcal H_c^k(n,p)$ is obtained by adding each $k$-subset of $[n]$ with probability $p$, and assigning it a color from $[c]$ uniformly, independently at random. As a first result we show that a typical $H\sim \mathcal H^2_c(n,p)$ (that is, a random edge colored graph) contains a rainbow Hamilton cycle, provided that $c=(1+o(1))n$ and $p=\frac{\log n+\log\log n+ω(1)}{n}$. This is asymptotically best possible with respect to both parameters, and improves a result of Frieze and Loh. Secondly, based on an ingenious coupling idea of McDiarmid, we provide a general tool for tackling problems related to finding "nicely edge colored" structures in random graphs/hypergraphs. We illustrate the generality of this statement by presenting two interesting applications. In one application we show that a typical $H\sim \mathcal H^k_c(n,p)$ contains a rainbow copy of a hypergraph $S$, provided that $c=(1+o(1))|E(S)|$ and $p$ is (up to a multiplicative constant) a threshold function for the property of containment of a copy of $S$. In the second application we show that a typical $G\sim \mathcal H_{c}^2(n,p)$ contains $(1-o(1))np/2$ edge disjoint Hamilton cycles, each of which is rainbow, provided that $c=ω(n)$ and $p=ω(\log n/n)$.

math.CO↗

arXiv2015

On saturation games

A graph $G = (V,E)$ is said to be saturated with respect to a monotone increasing graph property ${\mathcal P}$, if $G \notin {\mathcal P}$ but $G \cup \{e\} \in {\mathcal P}$ for every $e \in \binom{V}{2} \setminus E$. The saturation game $(n, {\mathcal P})$ is played as follows. Two players, called Mini and Max, progressively build a graph $G \subseteq K_n$, which does not satisfy ${\mathcal P}$. Starting with the empty graph on $n$ vertices, the two players take turns adding edges $e \in \binom{V(K_n)}{2} \setminus E(G)$, for which $G \cup \{e\} \notin {\mathcal P}$, until no such edge exists (i.e. until $G$ becomes ${\mathcal P}$-saturated), at which point the game is over. Max's goal is to maximize the length of the game, whereas Mini aims to minimize it. The score of the game, denoted by $s(n, {\mathcal P})$, is the number of edges in $G$ at the end of the game, assuming both players follow their optimal strategies. We prove lower and upper bounds on the score of games in which the property the players need to avoid is being $k$-connected, having chromatic number at least $k$, and admitting a matching of a given size. In doing so we demonstrate that the score of certain games can be as large as the Turán number or as low as the saturation number of the respective graph property. We also demonstrate that the score might strongly depend on the identity of the first player to move.

math.CO↗

arXiv2015

Counting and packing Hamilton $\ell$-cycles in dense hypergraphs

We consider problems about packing and counting Hamilton $\ell$-cycles in hypergraphs of large minimum degree. Given a hypergraph $\mathcal H$, for a $d$-subset $A\subseteq V(\mathcal H)$, we denote by $d_{\mathcal H}(A)$ the number of distinct \emph{edges} $f\in E(\mathcal H)$ for which $A\subseteq f$, and set $δ_d(\mathcal H)$ to be the minimum $d_{\mathcal H}(A)$ over all $A\subseteq V(\mathcal H)$ of size $d$. We show that if a $k$-uniform hypergraph on $n$ vertices $\mathcal H$ satisfies $δ_{k-1}(\mathcal H)\geq αn$ for some $α>1/2$, then for every $\ell 1/2$, then $\mathcal H$ contains $f(α)n$ edge-disjoint Hamilton $\ell$-cycles for an explicit function $f(α)>0$. For the case where every $(k-1)$-tuple $X\subset V({\mathcal H})$ satisfies $d_{\mathcal H}(X)\in (α\pm o(1))n$, we show that $\mathcal H$ contains edge-disjoint Haimlton $\ell$-cycles which cover all but $o\left(|E(\mathcal H)|\right)$ edges of $\mathcal H$. As a tool we prove the following result which might be of independent interest: For a bipartite graph $G$ with both parts of size $n$, with minimum degree at least $δn$, where $δ>1/2$, and for $p=ω(\log n/n)$ the following holds. If $G$ contains an $r$-factor for $r=Θ(n)$, then by retaining edges of $G$ with probability $p$ independently at random, w.h.p the resulting graph contains a $(1-o(1))rp$-factor.

math.CO↗

arXiv2014

Decomposing random graphs into few cycles and edges

Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random graph $G(n,p)$ with probability approaching 1 as $n\rightarrow\infty$. In this paper we show that for most edge probabilities $G(n,p)$ can be decomposed into a union of $\frac{n}{4}+\frac{np}{2}+o(n)$ cycles and edges whp. This result is asymptotically tight.

math.CO↗

arXiv2014

Large subgraphs without short cycles

We study two extremal problems about subgraphs excluding a family $\F$ of graphs. i) Among all graphs with $m$ edges, what is the smallest size $f(m,\F)$ of a largest $\F$--free subgraph? ii) Among all graphs with minimum degree $δ$ and maximum degree $Δ$, what is the smallest minimum degree $h(δ,Δ,\F)$ of a spanning $\F$--free subgraph with largest minimum degree? These questions are easy to answer for families not containing any bipartite graph. We study the case where $\F$ is composed of all even cycles of length at most $2r$, $r\geq 2$. In this case, we give bounds on $f(m,\F)$ and $h(δ,Δ,\F)$ that are essentially asymptotically tight up to a logarithmic factor. In particular for every graph $G$, we show the existence of subgraphs with arbitrarily high girth, and with either many edges or large minimum degree. These subgraphs are created using probabilistic embeddings of a graph into extremal graphs.

math.CO↗

arXiv2014

Compatible Hamilton cycles in Dirac graphs

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once. A celebrated theorem of Dirac from 1952 asserts that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper 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 $\mathcal{F}=\{F_v\}_{v\in V}$ such that 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 $\{e,e'\} \notin F_v$, where $v=e\cap e'$. This notion is partly motivated by a concept of transition systems defined by Kotzig in 1968, and can be viewed as a quantitative measure of robustness of graph properties. We prove that there is a constant $μ>0$ such that for every $μn$-bounded incompatibility system $\mathcal{F}$ over a Dirac graph $G$, there exists a Hamilton cycle compatible with $\mathcal{F}$. This settles in a very strong form, a conjecture of Häggkvist from 1988.

math.CO↗

arXiv2014

Efficient winning strategies in random-turn Maker-Breaker games

We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2007. A $p$-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is $p$). We analyze the random-turn version of several classical Maker-Breaker games such as the game Box (introduced by Chvátal and Erd\H os in 1987), the Hamilton cycle game and the $k$-vertex-connectivity game (both played on the edge set of $K_n$). For each of these games we provide each of the players with a (randomized) efficient strategy which typically ensures his win in the asymptotic order of the minimum value of $p$ for which he typically wins the game, assuming optimal strategies of both players.

math.CO↗

arXiv2014

Walker-Breaker games

We introduce and analyze the Walker-Breaker game, a variant of Maker-Breaker games where Maker is constrained to choose edges of a walk or path in a given graph G, with the goal of visiting as many vertices of the underlying graph as possible.

math.CO↗

arXiv2014

Positional Games

Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.

math.CO↗

arXiv2014

Packing tree factors in random and pseudo-random graphs

For a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on t vertices and ε> 0, the random graph G_{n,p}, with n a multiple of t, with high probability contains a family of edge-disjoint T-factors covering all but an ε-fraction of its edges, as long as ε^4 n p >> (log n)^2. Assuming stronger divisibility conditions, the edge probability can be taken down to p > (C log n)/n. A similar packing result is proved also for pseudo-random graphs, defined in terms of their degrees and co-degrees.

math.CO↗

arXiv2014

Cores of random graphs are born Hamiltonian

Let $(G_t)_{t \geq 0}$ be the random graph process ($G_0$ is edgeless and $G_t$ is obtained by adding a uniformly distributed new edge to $G_{t-1}$), and let $τ_k$ denote the minimum time $t$ such that the $k$-core of $G_t$ (its unique maximal subgraph with minimum degree at least $k$) is nonempty. For any fixed $k\geq 3$ the $k$-core is known to emerge via a discontinuous phase transition, where at time $t=τ_k$ its size jumps from 0 to linear in the number of vertices with high probability. It is believed that for any $k\geq 3$ the core is Hamiltonian upon creation w.h.p., and Bollobás, Cooper, Fenner and Frieze further conjectured that it in fact admits $\lfloor(k-1)/2\rfloor$ edge-disjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the $k$-core in $G(n,p)$ was unknown for any $k$. We show here that for any fixed $k\ge 15$ the $k$-core of $G_t$ is w.h.p. Hamiltonian for all $t \geq τ_k$, i.e., immediately as the $k$-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed $k$ the $k$-core contains $\lfloor (k-3)/2\rfloor$ edge-disjoint Hamilton cycles w.h.p. for all $t\geq τ_k$.

math.CO↗

arXiv2014

Contagious Sets in Expanders

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, where $r>1$ is the activation threshold. 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. Computing $m(G,r)$ is NP-hard. It is known that for every $d$-regular or nearly $d$-regular graph on $n$ vertices, $m(G,r) \le O(\frac{nr}{d})$. We consider such graphs that additionally have expansion properties, parameterized by the spectral gap and/or the girth of the graphs. The general flavor of our results is that sufficiently strong expansion (e.g., $λ(G)=O(\sqrt{d})$, or girth $Ω(\log \log d)$) implies that $m(G,2) \le O(\frac{n}{d^2})$ (and more generally, $m(G,r) \le O(\frac{n}{d^{r/(r-1)}})$). Significantly weaker expansion properties suffice in order to imply that $m(G,2)\le O(\frac{n \log d}{d^2})$. For example, we show this for graphs of girth at least~7, and for graphs with $λ(G)<(1-ε)d$, provided the graph has no 4-cycles. Nearly $d$-regular expander graphs can be obtained by considering the binomial random graph $G(n,p)$ with $p \simeq \frac{d}{n}$ and $d > \log n$. For such graphs we prove that $Ω(\frac{n}{d^2 \log d}) \le m(G,2) \le O(\frac{n\log\log d}{d^2\log d})$ almost surely. Our results are algorithmic, entailing simple and efficient algorithms for selecting contagious sets.

cs.DM↗

arXiv2014

Long paths and cycles in random subgraphs of H-free graphs

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $δ$ (depending only on $\varepsilon$) such that the following holds: If $p \ge \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(δk)$, where $n_{\mathcal{H}}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

math.CO↗

arXiv2013

Avoider-Enforcer games played on edge disjoint hypergraphs

We analyze Avoider-Enforcer games played on edge disjoint hypergraphs, providing an analog of the classic and well known game $Box$, due to Chvátal and Erdős. We consider both strict and monotone versions of Avoider-Enforcer games, and for each version we give a sufficient condition to win for each player. We also present applications of our results to several general Avoider-Enforcer games.

math.CO↗

arXiv2013

On the non-planarity of a random subgraph

Let $G$ be a finite graph with minimum degree $r$. Form a random subgraph $G_p$ of $G$ by taking each edge of $G$ into $G_p$ independently and with probability $p$. We prove that for any constant $ε>0$, if $p=\frac{1+ε}{r}$, then $G_p$ is non-planar with probability approaching 1 as $r$ grows. This generalizes classical results on planarity of binomial random graphs.

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