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Maksim Zhukovskii

Publications and source records attributed to Maksim Zhukovskii.

At least 37 records · Page 2Linked to original sources
arXiv2025

A large hole in pseudo-random graphs

We show that there exist constants $δ_1,δ_2>0$ such that if $G$ is an $(n,d,λ)$-graph with $λ/d\leδ_1$, then $G$ contains an induced cycle of length at least $δ_2n/d$. We further demonstrate that, up to a constant factor, this is best possible. Utilising our techniques, we derive that the number of non-isomorphic induced subgraphs of such $G$ is at least exponential in $n\log d/d$, and further demonstrate that this is tight up to a constant factor in the exponent.

math.CO↗
arXiv2025

Non-isomorphic subgraphs in random graphs

We establish the asymptotic behaviour of $μ(G(n,p))$, the number of unlabelled induced subgraphs in the binomial random graph $G(n,p)$, for almost the entire range of the probability parameter $p=p(n)\in[0,1]$. In particular, we show that typically the number of subgraphs becomes exponential when $p$ passes $1/n$, reaches maximum possible base of exponent (asymptotically) when $p\gg 1/n$, and reaches the asymptotic value $2^n$ when $p$ passes $2\ln n/n$. For $p\gg \ln n/n$, we get the first order term and asymptotics of the second order term of $μ(G(n,p))$. We also prove that random regular graphs $G_{n,d}$ typically have $μ(G_{n,d})\geq 2^{c_d n}$ for all $d\geq 3$ and some positive constant $c_d$ such that $c_d\to 1$ as $d\to\infty$.

math.CO↗
arXiv2025

Faces in girth-saturated graphs on surfaces

What is the maximum length ${\rm f}_{\rm max}(\ell, Σ)$ of a facial cycle of an inclusion-maximal graph with girth at least $\ell$ embedded on a given surface $Σ$? If $Σ=\mathcal{P}$ is a plane, we show that $3\ell-11\leq {\rm f}_{\rm max}(\ell, \mathcal{P})\leq 8\ell-13$. We also prove that ${\rm f}_{\rm max}(\ell, Σ)$ is bounded for any integer $\ell$ and any closed surface $Σ$. For a fixed $Σ$, we show that $Ω(\ell) ={\rm f}_{\rm max}(\ell, Σ) = O(\ell^2)$, while for a fixed $\ell\ge 6$, ${\rm f}_{\rm max}(\ell, Σ)=Θ(g)$, where $g$ is the genus of $Σ$.

math.CO↗
arXiv2025

Extremal independence in discrete random systems

Let $\mathbf{X}(n) \in \mathbb{R}^d$ be a sequence of random vectors, where $n\in\mathbb{N}$ and $d = d(n)$. Under certain weakly dependence conditions, we prove that the distribution of the maximal component of $\mathbf{X}$ and the distribution of the maximum of their independent copies are asymptotically equivalent. Our result on extremal independence relies on new lower and upper bounds for the probability that none of a given finite set of events occurs. As applications, we obtain the distribution of various extremal characteristics of random discrete structures such as maximum codegree in binomial random hypergraphs and the maximum number of cliques sharing a given vertex in binomial random graphs. We also generalise Berman-type conditions for a sequence of Gaussian random vectors to possess the extremal independence property.

math.PR↗
arXiv2025

Tree tilings in random regular graphs

We show that for every $ε>0$ there exists a sufficiently large $d_0\in \mathbb{N}$ such that for every $d\ge d_0$, whp the random $d$-regular graph $G(n,d)$ contains a $T$-factor for every tree $T$ on at most $(1-ε)d/\ln d$ vertices. This is best possible since, for large enough integer $d$, whp $G(n,d)$ does not contain a $\frac{(1+ε)d}{\ln d}$-star-factor. Our method gives a randomised algorithm which whp finds said $T$-factor and whose expected running time is $O(n^{1+o(1)})$, as well as an efficient deterministic counterpart.

math.CO↗
arXiv2024

Functionality of Random Graphs

The functionality of a graph $G$ is the minimum number $k$ such that in every induced subgraph of $G$ there exists a vertex whose neighbourhood is uniquely determined by the neighborhoods of at most $k$ other vertices in the subgraph. The functionality parameter was introduced in the context of adjacency labeling schemes, and it generalises a number of classical and recent graph parameters including degeneracy, twin-width, and symmetric difference. We establish the functionality of a random graph $G(n,p)$ up to a constant factor for every value of $p$.

math.CO↗
arXiv2024

On the maximum number of common neighbours in dense random regular graphs

We derive the distribution of the maximum number of common neighbours of a pair of vertices in a dense random regular graph.The proof involves two important steps. One step is to establish the extremal independence property: the asymptotic equivalence with the maximum component of a vector with independent marginal distributions. The other step is to prove that the distribution of the number of common neighbours for each pair of vertices can be approximated by the binomial distribution.

math.CO↗
arXiv2024

Gathering Information about a Graph by Counting Walks from a Single Vertex

We say that a vertex $v$ in a connected graph $G$ is decisive if the numbers of walks from $v$ of each length determine the graph $G$ rooted at $v$ up to isomorphism among all connected rooted graphs with the same number of vertices. On the other hand, $v$ is called ambivalent if it has the same walk counts as a vertex in a non-isomorphic connected graph with the same number of vertices as $G$. Using the classical constructions of cospectral trees, we first observe that ambivalent vertices exist in almost all trees. If a graph $G$ is determined by spectrum and its characteristic polynomial is irreducible, then we prove that all vertices of $G$ are decisive. Note that both assumptions are conjectured to be true for almost all graphs. Without using any assumption, we are able to prove that the vertices of a random graph are with high probability distinguishable from each other by the numbers of closed walks of length at most 4. As a consequence, the closed walk counts for lengths 2, 3, and 4 provide a canonical labeling of a random graph. Answering a question posed in chemical graph theory, we finally show that all walk counts for a vertex in an $n$-vertex graph are determined by the counts for the $2n$ shortest lengths, and the bound $2n$ is here asymptotically tight.

cs.DM↗
arXiv2024

Canonical labelling of sparse random graphs

We show that if $p=O(1/n)$, then the Erdős-Rényi random graph $G(n,p)$ with high probability admits a canonical labeling computable in time $O(n\log n)$. Combined with the previous results on the canonization of random graphs, this implies that $G(n,p)$ with high probability admits a polynomial-time canonical labeling whatever the edge probability function $p$. Our algorithm combines the standard color refinement routine with simple post-processing based on the classical linear-time tree canonization. Noteworthy, our analysis of how well color refinement performs in this setting allows us to complete the description of the automorphism group of the 2-core of $G(n,p)$.

cs.DM↗
arXiv2024

First order complexity of finite random structures

For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO convergence law correspond to FO complexities equal to $\{0,1\}$ and a subset of $\mathbb{R}$, respectively. We present a hierarchy of FO complexity classes, introduce a stochastic FO reduction that allows to transfer complexity results between different random structures, and deduce using this tool several new logical limit laws for binomial random structures. Finally, we introduce a conditional distribution on graphs, subject to a FO sentence $φ$, that generalises certain well-known random graph models, show instances of this distribution for every complexity class, and prove that the set of all $φ$ validating 0--1 law is not recursively enumerable.

cs.LO↗
arXiv2024

Maximum induced trees and forests of bounded degree in random graphs

Asymptotic behaviour of maximum sizes of induced trees and forests has been studied extensively in last decades, though the overall picture is far from being complete. In this paper, we close several significant gaps: 1) We prove $2$-point concentration of the maximum sizes of an induced forest and an induced tree with maximum degree at most $Δ$ in dense binomial random graphs $G(n,p)$ with constant probability $p$. 2) We show concentration in an explicit interval of size $o(1/p)$ for the maximum size of an induced forest with maximum degree at most $Δ$ for $1/n\ll p=o(1)$. Our proofs rely on both the second moment approach, with the probabilistic part involving Talagrand's concentration inequality and the analytical part involving saddle-point analysis, and new results on enumeration of labelled trees and forests that might be of their own interest.

math.CO↗
arXiv2024

Reconstructing random graphs from distance queries

We estimate the minimum number of distance queries that is sufficient to reconstruct the binomial random graph $G(n,p)$ with constant diameter with high probability. We get a tight (up to a constant factor) answer for all $p>n^{-1+o(1)}$ outside "threshold windows" around $n^{-k/(k+1)+o(1)}$, $k\in\mathbb{Z}_{>0}$: with high probability the query complexity equals $Θ(n^{4-d}p^{2-d})$, where $d$ is the diameter of the random graph. This demonstrates the following non-monotone behaviour: the query complexity jumps down at moments when the diameter gets larger; yet, between these moments the query complexity grows. We also show that there exists a non-adaptive algorithm that reconstructs the random graph with $O(n^{4-d}p^{2-d}\ln n)$ distance queries with high probability, and this is best possible.

math.CO↗
arXiv2024

Sprinkling with random regular graphs

We conjecture that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of the combined degree, provided it grows as the number of vertices tends to infinity. We verify this conjecture for the cases when the graphs are sufficiently dense or sparse. We also prove an asymptotic formula for the expected number of spanning regular subgraphs in a random regular graph.

math.CO↗
arXiv2024

Weak saturation rank: a failure of linear algebraic approach to weak saturation

Given a graph $F$ and a positive integer $n$, the weak $F$-saturation number $\mathrm{wsat}(K_n,F)$ is the minimum number of edges in a graph $H$ on $n$ vertices such that the edges missing in $H$ can be added, one at a time, so that every edge creates a copy of $F$. Kalai in 1985 introduced a linear algebraic approach that became one of the most efficient tools to prove lower bounds on weak saturation numbers. If $W$ is a vector space spanned by vectors $w(e)$ assigned to edges $e$ of $K_n$ in such a way that, for every copy $F'\subset K_n$ of $F$, there exist non-zero $λ_e$, $e\in E(F')$, satisfying $\sum_{e\in E(F')}λ_e w(e)=0$, then $\mathrm{dim}W\leq \mathrm{wsat}(K_n,F)$. In this paper, we prove limitations of this approach: we show infinitely many $F$ such that, for every vector space $W$ as above, $\mathrm{dim}W<\mathrm{wsat}(K_n,F)$. We also suggest a modification of this approach that allows to get tight lower bounds even when the original linear algebraic approach is not sufficient. Finally, we generalise our results to random graphs, complete multipartite graphs, and hypergraphs.

math.CO↗
arXiv2024

First order distinguishability of sparse random graphs

We study the problem of distinguishing between two independent samples $\mathbf{G}_n^1,\mathbf{G}_n^2$ of a binomial random graph $G(n,p)$ by first order (FO) sentences. Shelah and Spencer proved that, for a constant $α\in(0,1)$, $G(n,n^{-α})$ obeys FO zero-one law if and only if $α$ is irrational. Therefore, for irrational $α\in(0,1)$, any fixed FO sentence does not distinguish between $\mathbf{G}_n^1,\mathbf{G}_n^2$ with asymptotical probability 1 (w.h.p.) as $n\to\infty$. We show that the minimum quantifier depth $\mathbf{k}_α$ of a FO sentence $φ=φ(\mathbf{G}_n^1,\mathbf{G}_n^2)$ distinguishing between $\mathbf{G}_n^1,\mathbf{G}_n^2$ depends on how closely $α$ can be approximated by rationals: (1) for all non-Liouville $α\in(0,1)$, $\mathbf{k}_α=Ω(\ln\ln\ln n)$ w.h.p.; (2) there are irrational $α\in(0,1)$ with $\mathbf{k}_α$ that grow arbitrarily slowly w.h.p.; (3) $\mathbf{k}_α=O_p(\frac{\ln n}{\ln\ln n})$ for all $α\in(0,1)$. The main ingredients in our proofs are a novel randomized algorithm that generates asymmetric strictly balanced graphs as well as a new method to study symmetry groups of randomly perturbed graphs.

math.CO↗
arXiv2024

A Jump of the Saturation Number in Random Graphs?

For graphs $G$ and $F$, the saturation number $\textit{sat}(G,F)$ is the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$. In 2017, Korándi and Sudakov initiated the study of saturation in random graphs. They showed that for constant $p\in (0,1)$, whp $\textit{sat}\left(G(n,p),K_s\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$. We show that for every graph $F$ and every constant $p\in (0,1)$, whp $\textit{sat}\left(G(n,p), F\right)=O(n\ln n)$. Furthermore, if every edge of $F$ belongs to a triangle, then the above is the right asymptotic order of magnitude, that is, whp $\textit{sat}\left(G(n,p),F\right)=Θ(n\ln n)$. We further show that for a large family of graphs $\mathcal{F}$ with an edge that does not belong to a triangle, which includes all the bipartite graphs, for every $F\in \mathcal{F}$ and constant $p\in(0,1)$, whp $\textit{sat}\left(G(n,p),F\right)=O(n)$. We conjecture that this sharp transition from $O(n)$ to $Θ(n\ln n)$ depends only on this property, that is, that for any graph $F$ with at least one edge that does not belong to a triangle, whp $\textit{sat}\left(G(n,p),F\right)=O(n)$. We further generalise the result of Korándi and Sudakov, and show that for a more general family of graphs $\mathcal{F}'$, including all complete graphs $K_s$ and all complete multipartite graphs of the form $K_{1,1,s_3,\ldots, s_{\ell}}$, for every $F\in \mathcal{F}'$ and every constant $p\in(0,1)$, whp $\textit{sat}\left(G(n,p),F\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$. Finally, we show that for every complete multipartite graph $K_{s_1, s_2, \ldots, s_{\ell}}$ and every $p\in \left[\frac{1}{2},1\right)$, $\textit{sat}\left(G(n,p),K_{s_1,s_2,\ldots,s_{\ell}}\right)=\left(1+o(1)\right)n\log_{\frac{1}{1-p}}n$.

math.CO↗
arXiv2024

Stability of large cuts in random graphs

We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-Ω(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the same manner by all maximum cuts of $G_{n,p}$. Moreover, the analogous statement remains true when one replaces maximum cuts with nearly-maximum cuts. We then demonstrate how one can use this statement as a tool for showing that certain properties of $G_{n,p}$ that hold in a fixed balanced cut hold simultaneously in all maximum cuts. We provide two example applications of this tool. First, we prove that maximum cuts in $G_{n,p}$ typically partition the neighbourhood of every vertex into nearly equal parts; this resolves a conjecture of DeMarco and Kahn for all but a narrow range of densities $p$. Second, for all edge-critical, nonbipartite, and strictly 2-balanced graphs $H$, we prove a lower bound on the threshold density $p$ above which every largest $H$-free subgraph of $G_{n,p}$ is $(χ(H)-1)$-partite. Our lower bound exactly matches the upper bound on this threshold recently obtained by the first two authors.

math.CO↗