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Maksim Zhukovskii

Publications and source records attributed to Maksim Zhukovskii.

At least 73 records · Page 4Linked to original sources
arXiv2022

Distribution of tree parameters by martingale approach

For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem and the Aldous--Broder algorithm. In particular, our general result implies the asymptotic normality of the number of occurrences of any given small pattern and the asymptotic log-normality of the number of automorphisms.

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arXiv2022

Cycle saturation in random graphs

For a fixed graph $F,$ the minimum number of edges in an edge-maximal $F$-free subgraph of $G$ is called the $F$-saturation number. The asymptotics of the $F$-saturation number of the binomial random graph $G(n,p)$ for constant $p\in(0,1)$ is known for complete graphs $F=K_m$ and stars $F=K_{1,m}.$ This paper is devoted to the case when the pattern graph $F$ is a simple cycle $C_m.$ We prove that, for $m\geqslant 5,$ whp $\mathrm{sat}\left(G\left(n,p\right),C_m\right) = n+Θ\left(\frac{n}{\ln n}\right).$ Also we find $c=c(p)$ such that whp $\frac{3}{2}n(1+o(1))\leqslant\mathrm{sat}\left(G\left(n,p\right),C_4\right)\leqslant cn(1+o(1)).$ In particular, whp $\mathrm{sat}\left(G\left(n,\frac{1}{2}\right),C_4\right)\leqslant\frac{27}{14}n(1+o(1)).$

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arXiv2022

Large cycles in generalized Johnson graphs

We count cycles of an unbounded length in generalized Johnson graphs. Asymptotics of the number of such cycles is obtained for certain growth rates of the cycle length.

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arXiv2022

EMSO(FO$^2$) 0-1 law fails for all dense random graphs

In this paper, we disprove EMSO(FO$^2$) convergence law for the binomial random graph $G(n,p)$ for any constant probability $p$. More specifically, we prove that there exists an existential monadic second order sentence with 2 first order variables such that, for every $p\in(0,1)$, the probability that it is true on $G(n,p)$ does not converge.

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arXiv2021

$γ$-variable first-order logic of uniform attachment random graphs

We study logical limit laws for uniform attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time $n+1$, the vertex $n+1$ is introduced together with $m$ edges joining the new vertex with $m$ different vertices chosen uniformly at random from $1,\ldots,n$. We prove that this random graph obeys convergence law for first-order sentences with at most $m-2$ variables.

math.PR↗
arXiv2021

On the sizes of large subgraphs of the binomial random graph

We consider the binomial random graph $G(n,p)$, where $p$ is a constant, and answer the following two questions. First, given $e(k)=p{k\choose 2}+O(k)$, what is the maximum $k$ such that a.a.s.~the binomial random graph $G(n,p)$ has an induced subgraph with $k$ vertices and $e(k)$ edges? We prove that this maximum is not concentrated in any finite set (in contrast to the case of a small $e(k)$). Moreover, for every constant $C>0$ and every $ω_n\to\infty$, a.a.s.~the size of the concentration set belongs to $(C\sqrt{n/\ln n},ω_n\sqrt{n/\ln n})$. Second, given $k>\varepsilon n$, what is the maximum $μ$ such that a.a.s.~the set of sizes of $k$-vertex subgraphs of $G(n,p)$ contains a full interval of length $μ$? The answer is $μ=Θ\left(\sqrt{(n-k)n\ln{n\choose k}}\right)$.

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arXiv2021

On monotonicity of Ramanujan function for binomial random variables

For a binomial random variable $ξ$ with parameters $n$ and $b/n$, it is well known that the median equals $b$ when $b$ is an integer. In 1968, Jogdeo and Samuels studied the behaviour of the relative difference between ${\sf P}(ξ=b)$ and $1/2-{\sf P}(ξ<b)$. They proved its monotonicity in $n$ and posed a question about its monotonicity in $b$. This question is motivated by the solved problem proposed by Ramanujan in 1911 on the monotonicity of the same quantity but for a Poisson random variable with an integer parameter $b$. In the paper, we answer this question and introduce a simple way to analyse the monotonicity of similar functions.

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arXiv2021

Large cycles in random generalized Johnson graphs

This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.

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arXiv2020

Zero-one laws for existential first order sentences of bounded quantifier depth

For any fixed positive integer $k$, let $α_{k}$ denote the smallest $α\in (0,1)$ such that the random graph sequence $\left\{G\left(n, n^{-α}\right)\right\}$ does not satisfy the zero-one law for the set $\mathcal{E}_{k}$ of all existential first order sentences that are of quantifier depth at most $k$. This paper finds upper and lower bounds on $α_{k}$, showing that as $k \rightarrow \infty$, we have $α_{k} = \left(k - 2 - t(k)\right)^{-1}$ for some function $t(k) = Θ(k^{-2})$. We also establish the precise value of $α_{k}$ when $k = 4$.

math.PR↗
arXiv2019

Maximum sparse induced subgraphs of the binomial random graph with given number of edges

We prove that a.a.s. the maximum size of an induced subtree of the binomial random graph $G(n,p)$ is concentrated in 2 consecutive points. We also prove that, given a non-negative integer-valued function $t(k)<\varepsilon k^2$, under a certain smoothness condition on this function, a.a.s. the maximum size $k$ of an induced subgraph with exactly $t(k)$ edges of $G(n,p)$ is concentrated in 2 consecutive points as well.

math.CO↗
arXiv2019

Logical laws for short existential monadic second order sentences about graphs

In this paper, we study existential monadic second order (EMSO) properties of undirected graphs. In 2001, J.-M. Le Bars proved that there exists an EMSO sentence about undirected graphs such that the probability that it is true does not converge (here, the probability distribution is uniform over the set of all graphs on the fixed set of vertices). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the 0-1 law holds (every sentence is either true asymptotically almost surely (a.a.s.), or false a.a.s.). We give an example of EMSO sentence with 1 monadic variable without convergence and an example of EMSO sentence with 3 first order variables without convergence.

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arXiv2019

On the First-Order Complexity of Induced Subgraph Isomorphism

Given a graph $F$, let $I(F)$ be the class of graphs containing $F$ as an induced subgraph. Let $W[F]$ denote the minimum $k$ such that $I(F)$ is definable in $k$-variable first-order logic. The recognition problem of $I(F)$, known as Induced Subgraph Isomorphism (for the pattern graph $F$), is solvable in time $O(n^{W[F]})$. Motivated by this fact, we are interested in determining or estimating the value of $W[F]$. Using Olariu's characterization of paw-free graphs, we show that $I(K_3+e)$ is definable by a first-order sentence of quantifier depth 3, where $K_3+e$ denotes the paw graph. This provides an example of a graph $F$ with $W[F]$ strictly less than the number of vertices in $F$. On the other hand, we prove that $W[F]=4$ for all $F$ on 4 vertices except the paw graph and its complement. If $F$ is a graph on $t$ vertices, we prove a general lower bound $W[F]>(1/2-o(1))t$, where the function in the little-o notation approaches 0 as $t$ inreases. This bound holds true even for a related parameter $W^*[F]\le W[F]$, which is defined as the minimum $k$ such that $I(F)$ is definable in the infinitary logic $L^k_{\inftyω}$. We show that $W^*[F]$ can be strictly less than $W[F]$. Specifically, $W^*[P_4]=3$ for $P_4$ being the path graph on 4 vertices. Using the lower bound for $W[F]$, we also obtain a succintness result for existential monadic second-order logic: A usage of just one monadic quantifier sometimes reduces the first-order quantifier depth at a super-recursive rate.

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arXiv2018

Existential monadic second order logic of undirected graphs: a disproof of the Le Bars conjecture

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) about undirected graphs. He proved that there exists an EMSO sentence $ϕ$ such that ${\sf P}(G_n \modelsϕ)$ does not converge as $n\to\infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1, \dots, n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.

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arXiv2018

Short monadic second order sentences about sparse random graphs

In this paper, we study zero-one laws for the Erdős--Rényi random graph model $G(n,p)$ in the case when $p = n^{-α}$ for $α>0$. For a given class $\mathcal{K}$ of logical sentences about graphs and a given function $p=p(n)$, we say that $G(n,p)$ obeys the zero-one law (w.r.t. the class $\mathcal{K}$) if each sentence $φ\in\mathcal{K}$ either a.a.s. true or a.a.s. false for $G(n,p)$. In this paper, we consider first order properties and monadic second order properties of bounded \textit{quantifier depth} $k$, that is, the length of the longest chain of nested quantifiers in the formula expressing the property. Zero-one laws for properties of quantifier depth $k$ we call the \textit{zero-one $k$-laws}. The main results of this paper concern the zero-one $k$-laws for monadic second order properties (MSO properties). We determine all values $α>0$, for which the zero-one $3$-law for MSO properties does not hold. We also show that, in contrast to the case of the $3$-law, there are infinitely many values of $α$ for which the zero-one $4$-law for MSO properties does not hold. To this end, we analyze the evolution of certain properties of $G(n,p)$ that may be of independent interest.

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arXiv2018

Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

Let $v(F)$ denote the number of vertices in a fixed connected pattern graph $F$. We show an infinite family of patterns $F$ such that the existence of a subgraph isomorphic to $F$ is expressible by a first-order sentence of quantifier depth $\frac23\,v(F)+1$, assuming that the host graph is sufficiently large and connected. On the other hand, this is impossible for any $F$ with using less than $\frac23\,v(F)-2$ first-order variables.

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arXiv2017

First order sentences about random graphs: small number of alternations

Spectrum of a first order sentence is the set of all $α$ such that $G(n, n^{-α})$ does not obey zero-one law w.r.t. this sentence. We have proved that the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals 3. We have also proved that the spectrum of a first order sentence with a quantifier depth 4 has no limit points except possibly the points 1/2 and 3/5.

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