arXiv subjects

Gal Kronenberg

Publications and source records attributed to Gal Kronenberg.

25 records · Page 2Linked to original sources

arXiv2018

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $χ(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.

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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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arXiv2016

Optimal Threshold for a Random Graph to be 2-Universal

For a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-universal if $G$ contains every graph in $\mathcal{F}$ as a (not necessarily induced) subgraph. For the family of all graphs on $n$ vertices and of maximum degree at most two, $\mathcal{H}(n,2)$, we prove that there exists a constant $C$ such that for $p \geq C \left( \frac{\log n}{n^2} \right)^{\frac{1}{3}}$, the binomial random graph $G(n,p)$ is typically $\mathcal{H}(n,2)$-universal. This bound is optimal up to the constant factor as illustrated in the seminal work of Johansson, Kahn, and Vu for triangle factors. Our result improves significantly on the previous best bound of $p \geq C \left(\frac{\log n}{n}\right)^{\frac{1}{2}}$ due to Kim and Lee. In fact, we prove the stronger result that for the family of all graphs on $n$ vertices, of maximum degree at most two and of girth at least $\ell$, $\mathcal{H}^{\ell}(n,2)$, $G(n,p)$ is typically $\mathcal H^{\ell}(n,2)$-universal when $p \geq C \left(\frac{\log n}{n^{\ell -1}}\right)^{\frac{1}{\ell}}$. This result is also optimal up to the constant factor. Our results verify (in a weak form) a classical conjecture of Kahn and Kalai.

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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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arXiv2015

Packing, Counting and Covering Hamilton cycles in random directed graphs

A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Pos?a `rotation-extension' technique for the undirected analogue. Let ${\mathcal D}(n,p)$ denote the random digraph on vertex set $[n]$, obtained by adding each directed edge independently with probability $p$. Here, we present a general and a very simple method, using known results, to attack problems of packing and counting Hamilton cycles in random directed graphs, for every edge-probability $p>\log^C(n)/n$. Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case.

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arXiv2014

Packing a randomly edge-colored random graph with rainbow $k$-outs

Let $G$ be a graph on $n$ vertices and let $k$ be a fixed positive integer. We denote by $\mathcal G_{\text{$k$-out}}(G)$ the probability space consisting of subgraphs of $G$ where each vertex $v\in V(G)$ randomly picks $k$ neighbors from $G$, independently from all other vertices. We show that if $δ(G)=ω(\log n)$ and $k\geq 2$, then the following holds for every $p=ω(\log n/δ(G))$. Let $H$ be a random graph obtained by keeping each $e\in E(G)$ with probability $p$ independently at random and then coloring its edges independently and uniformly at random with elements from the set $[kn]$. Then, w.h.p. $H$ contains $t:=(1-o(1))δ(G)p/(2k)$ edge-disjoint graphs $H_1,...,H_t$ such that each of the $H_i$ is \emph{rainbow} (that is, all the edges are colored with distinct colors), and such that for every monotone increasing property of graphs $\mathcal P$ and for every $1\leq i\leq t$ we have $\Pr[\mathcal G_{\text{$k$-out}}(G)\models \mathcal P]\leq \Pr[H_i\models \mathcal P]+n^{-ω(1)}$. Note that since (in this case) a typical member of $\mathcal G_{\text{$k$-out}}(G)$ has average degree roughly $2k$, this result is asymptotically best possible. We present several applications of this; for example, we use this result to prove that for $p=ω(\log n/n)$ and $c=23n$, a graph $H\sim \mathcal G_{c}(K_n,p)$ w.h.p. contains $(1-o(1))np/46$ edge-disjoint rainbow Hamilton cycles. More generally, using a recent result of Frieze and Johansson, the same method allows us to prove that if $G$ has minimum degree $δ(G)\geq (1+\varepsilon)n/2$, then there exist functions $c=O(n)$ and $t=Θ(np)$ (depending on $\varepsilon$) such that the random subgraph $H\sim \mathcal G_{c}(G,p)$ w.h.p. contains $t$ edge-disjoint rainbow Hamilton cycles.

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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.

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