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Maksim Zhukovskii

Publications and source records attributed to Maksim Zhukovskii.

At least 55 records · Page 3Linked to original sources
arXiv2024

Tight bounds on adjacency labels for monotone graph classes

A class of graphs admits an adjacency labeling scheme of size $b(n)$, if the vertices in each of its $n$-vertex graphs can be assigned binary strings (called labels) of length $b(n)$ so that the adjacency of two vertices can be determined solely from their labels. We give tight bounds on the size of adjacency labels for every family of monotone (i.e., subgraph-closed) classes with a well-behaved growth function between $2^{O(n \log n)}$ and $2^{O(n^{2-δ})}$ for any $δ> 0$. Specifically, we show that for any function $f: \mathbb N \to \mathbb R$ satisfying $\log n \leqslant f(n) \leqslant n^{1-δ}$ for any fixed $δ> 0$, and some~sub-multiplicativity condition, there are monotone graph classes with growth $2^{O(nf(n))}$ that do not admit adjacency labels of size at most $f(n) \log n$. On the other hand, any such class does admit adjacency labels of size $O(f(n)\log n)$. Surprisingly this tight bound is a $Θ(\log n)$ factor away from the information-theoretic bound of $Ω(f(n))$. The special case when $f = \log$ implies that the recently-refuted Implicit Graph Conjecture [Hatami and Hatami, FOCS 2022] also fails within monotone classes. We further show that the Implicit Graph Conjecture holds for all monotone \emph{small} classes. In other words, any monotone class with growth rate at most $n!\,c^n$ for some constant $c>0$, admits adjacency labels of information-theoretic order optimal size. In fact, we show a more general result that is of independent interest: any monotone small class of graphs has bounded degeneracy.We conjecture that the Implicit Graph Conjecture holds for all hereditary small classes.

math.CO↗
arXiv2024

Small But Unwieldy: A Lower Bound on Adjacency Labels for Small Classes

We show that for any natural number $s$, there is a constant $γ$ and a subgraph-closed class having, for any natural $n$, at most $γ^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size at most $s \log n$. In other words, for every $s$, there is a small (even tiny) monotone class without universal graphs of size $n^s$. Prior to this result, it was not excluded that every small class has an almost linear universal graph, or equivalently a labeling scheme with labels of size $(1+o(1))\log n$. The existence of such a labeling scheme, a scaled-down version of the recently disproved Implicit Graph Conjecture, was repeatedly raised [Gavoille and Labourel, ESA '07; Dujmović et al., JACM '21; Bonamy et al., SIDMA '22; Bonnet et al., Comb. Theory '22]. Furthermore, our small monotone classes have unbounded twin-width, thus simultaneously disprove the already-refuted Small conjecture; but this time with a self-contained proof, not relying on elaborate group-theoretic constructions.

math.CO↗
arXiv2024

Spectrum of FO logic with quantifier depth 4 is finite

The $k$-spectrum is the set of all $α>0$ such that $G(n,n^{-α})$ does not obey the 0-1 law for FO sentences with quantifier depth at most $k$. In this paper, we prove that the minimum $k$ such that the $k$-spectrum is infinite equals 5.

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arXiv2023

Reconstruction of graph colourings

A $k$-deck of a (coloured) graph is a multiset of its induced $k$-vertex subgraphs. Given a graph $G$, when is it possible to reconstruct with high probability a uniformly random colouring of its vertices in $r$ colours from its $k$-deck? In this paper, we study this question for grids and random graphs. Reconstruction of random colourings of $d$-dimensional $n$-grids from the deck of their $k$-subgrids is one of the most studied colour reconstruction questions. The 1-dimensional case is motivated by the problem of reconstructing DNA sequences from their `shotgunned' stretches. It was comprehensively studied and the above reconstruction question was completely answered in the '90s. In this paper, we get a very precise answer for higher $d$. For every $d\geq 2$ and every $r\geq 2$, we present an almost linear algorithm that reconstructs with high probability a random $r$-colouring of vertices of a $d$-dimensional $n$-grid from the deck of all its $k$-subgrids for every $k\geq(d\log_r n)^{1/d}+1/d+\varepsilon$ and prove that the random $r$-colouring is not reconstructible with high probability if $k\leq (d\log_r n)^{1/d}-\varepsilon$. This answers the question of Narayanan and Yap (that was asked for $d\geq 3$) on "two-point concentration" of the minimum $k$ so that $k$-subgrids determine the entire colouring. Next, we prove that with high probability a uniformly random $r$-colouring of vertices of a uniformly random graph $G(n,1/2)$ is reconstructible from its full $k$-deck if $k\geq 2\log_2 n+8$ and is not reconstructible with high probability if $k\leq\sqrt{2\log_2 n}$. We further show that the colour reconstruction algorithm for random graphs can be modified and used for graph reconstruction: we prove that with high probability $G(n,1/2)$ is reconstructible from its full $k$-deck if $k\geq 2\log_2 n+11$ while it is not reconstructible with high probability if $k\leq 2\sqrt{\log_2 n}$.

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arXiv2023

On vertex Ramsey graphs with forbidden subgraphs

A classical vertex Ramsey result due to Nešetřil and Rödl states that given a finite family of graphs $\mathcal{F}$, a graph $A$ and a positive integer $r$, if every graph $B\in\mathcal{F}$ has a $2$-vertex-connected subgraph which is not a subgraph of $A$, then there exists an $\mathcal{F}$-free graph which is vertex $r$-Ramsey with respect to $A$. We prove that this sufficient condition for the existence of an $\mathcal{F}$-free graph which is vertex $r$-Ramsey with respect to $A$ is also necessary for large enough number of colours $r$. We further show a generalisation of the result to a family of graphs and the typical existence of such a subgraph in a dense binomial random graph.

math.CO↗
arXiv2023

Logical convergence laws via stochastic approximation and Markov processes

Since the paper of Kleinberg and Kleinberg, SODA'05, where it was proven that the preferential attachment random graph with degeneracy at least 3 does not obey the first order 0-1 law, no general methods were developed to study logical limit laws for recursive random graph models with arbitrary degeneracy. Even in the (possibly) simplest case of the uniform attachment, it is still not known whether the first order convergence law holds in this model. We prove that the uniform attachment random graph with bounded degrees obeys the first order convergence law. To prove the law, we describe dynamics of first order equivalence classes of the random graph using Markov chains. The convergence law follows from the existence of a limit distribution of the considered Markov chain. To show the latter convergence, we use stochastic approximation processes.

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arXiv2023

Maximum number of symmetric extensions in the random graph

It is known that after an appropriate rescaling the maximum degree of the binomial random graph converges in distribution to a Gumbel random variable. The same holds true for the maximum number of common neighbours of a $k$-vertex set, and for the maximum number of $s$-cliques sharing a single vertex. Can these results be generalised to the maximum number of extensions of a $k$-vertex set for any given way of extending of a $k$-vertex set by an $s$-vertex set? In this paper, we generalise the above mentioned results to a class of ``symmetric extensions'' and show that the limit distribution is not necessarily from the Gumbel family.

math.CO↗
arXiv2023

Weak saturation in graphs: a combinatorial approach

The weak saturation number $\mathrm{wsat}(n,F)$ is the minimum number of edges in a graph on $n$ vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of $F$. A usual approach to prove a lower bound for the weak saturation number is algebraic: if it is possible to embed edges of $K_n$ in a vector space in a certain way (depending on $F$), then the dimension of the subspace spanned by the images of the edges of $K_n$ is a lower bound for the weak saturation number. In this paper, we present a new combinatorial approach to prove lower bounds for weak saturation numbers that allows to establish worst-case tight (up to constant additive terms) general lower bounds as well as to get exact values of the weak saturation numbers for certain graph families. It is known (Alon, 1985) that, for every $F$, there exists $c_F$ such that $\mathrm{wsat}(n,F)=c_Fn(1+o(1))$. Our lower bounds imply that all values in the interval $\left[\fracδ{2}-\frac{1}{δ+1},δ-1\right]$ with step size $\frac{1}{δ+1}$ are achievable by $c_F$ (while any value outside this interval is not achievable).

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arXiv2023

Randomly twisted hypercubes -- between structure and randomness

Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these graphs have optimal diameter. We study twisted hypercubes in the setting where the instances can have general dependence, and also in the particular case where they are identical. We show that the resultant graph shares properties with random regular graphs, including small diameter, large vertex expansion, a semicircle law for its eigenvalues and no non-trivial automorphisms. However, in contrast to random regular graphs, twisted hypercubes allow for short routing schemes.

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arXiv2023

Saturation numbers of bipartite graphs in random graphs

For a given graph $F$, the $F$-saturation number of a graph $G$, denoted by $ {sat}(G, F)$, is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. In 2017, Korándi and Sudakov determined $ {sat}({G}(n, p), K_r)$ asymptotically, where ${G}(n, p) $ denotes the Erdős-Rényi random graph and $ K_r$ is the complete graph on $r$ vertices. In this paper, among other results, we present an asymptotic upper bound on ${sat}({G}(n, p), F)$ for any bipartite graph $F$ and also an asymptotic lower bound on ${sat}({G}(n, p), F)$ for any complete bipartite graph $F$.

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arXiv2023

On anti-stochastic properties of unlabeled graphs

We study vulnerability of a uniformly distributed random graph to an attack by an adversary who aims for a global change of the distribution while being able to make only a local change in the graph. We call a graph property $A$ anti-stochastic if the probability that a random graph $G$ satisfies $A$ is small but, with high probability, there is a small perturbation transforming $G$ into a graph satisfying $A$. While for labeled graphs such properties are easy to obtain from binary covering codes, the existence of anti-stochastic properties for unlabeled graphs is not so evident. If an admissible perturbation is either the addition or the deletion of one edge, we exhibit an anti-stochastic property that is satisfied by a random unlabeled graph of order $n$ with probability $(2+o(1))/n^2$, which is as small as possible. We also express another anti-stochastic property in terms of the degree sequence of a graph. This property has probability $(2+o(1))/(n\ln n)$, which is optimal up to factor of 2.

cs.DM↗
arXiv2023

Sharp thresholds for spanning regular graphs

Let $d\geq 3$ be a constant and let $F$ be a $d$-regular graph on $[n]$ with not too many symmetries. By the union bound, the probability threshold for the existence of a spanning subgraph in $G(n,p)$ isomorphic to $F$ is at least $p^*(n)=(1+o(1))(e/n)^{2/d}$. We give a tight bound on the edge expansion of $F$ guaranteeing that the probability threshold for the appearance of a copy of $F$ has the same order of magnitude as $p^*$. We also prove that, within a slight strengthening of this bound, the probability threshold is asymptotically equal to $p^*$. In particular, it proves the conjecture of Kahn, Narayanan and Park on a sharp threshold for the containment of a square of a Hamilton cycle. It also implies that, for $d\geq 4$ and (asymptotically) almost all $d$-regular graphs $F$ on $[n]$, $p(n)=(e/n)^{2/d}$ is a sharp threshold for $F$-containment.

math.CO↗
arXiv2023

Global information from local observations of the noisy voter model on a graph

We observe the outcome of the discrete time noisy voter model at a single vertex of a graph. We show that certain pairs of graphs can be distinguished by the frequency of repetitions in the sequence of observations. We prove that this statistic is asymptotically normal and that it distinguishes between (asymptotically) almost all pairs of finite graphs. We conjecture that the noisy voter model distinguishes between any two graphs other than stars.

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arXiv2022

Tight concentration of star saturation number in random graphs

For given graphs $F$ and $G$, the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$ is called the $F$-saturation number and denoted $\mathrm{sat}(G, F)$. For the star $F=K_{1,r}$, the asymptotics of $\mathrm{sat}(G(n,p),F)$ is known. We prove a sharper result: whp $\mathrm{sat}(G(n,p), K_{1,r})$ is concentrated in a set of 2 consecutive points.

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arXiv2022

Tight asymptotics of clique-chromatic numbers of dense random graphs

The clique chromatic number of a graph is the minimum number of colors required to assign to its vertex set so that no inclusion maximal clique is monochromatic. McDiarmid, Mitsche and Prałat proved that the clique chromatic number of the binomial random graph $G\left(n,\frac{1}{2}\right) $ is at most $\left(\frac{1}{2}+o(1)\right)\log_2n$ with high probability. Alon and Krivelevich showed that it is greater than $\frac{1}{2000}\log_2n$ with high probability and suggested that the right constant in front of the logarithm is $\frac{1}{2}.$ We prove their conjecture and, beyond that, obtain a tight concentration result: whp $χ_c\left(G\left(n,1/2\right)\right) = \frac{1}{2}\log_2 n - Θ\left(\ln\ln n\right).$

math.CO↗
arXiv2022

Covering three-tori with cubes

Let $μ(\varepsilon)$ be the minimum number of cubes of side $\varepsilon$ needed to cover the unit three-torus $[\mathbb{R}/\mathbb{Z}]^3$. We prove new lower and upper bounds for $μ(\varepsilon)$ and find the exact value for all $\varepsilon\geq\frac{7}{15}$ and all $\varepsilon\in\left[\frac{1}{r+1/(r^2+r+1)},\frac{1}{r-1/(r^2-1)}\right)$ for any integer $r\geq 2$.

math.CO↗
arXiv2022

Weak saturation stability

The paper studies wsat$(G,H)$ which is the minimum number of edges in a weakly $H$-saturated subgraph of $G$. We prove that wsat$(K_n,H)$ is `stable' - remains the same after independent removal of every edge of $K_n$ with constant probability - for all pattern graphs $H$ such that there exists a `local' set of edges percolating in $K_n$. This is true, for example, for cliques and complete bipartite graphs. We also find a threshold probability for the weak $K_{1,t}$-saturation stability.

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