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Lyuben Lichev

Publications and source records attributed to Lyuben Lichev.

At least 19 recordsLinked to original sources
arXiv2026

Consensus in label propagation on binomial random graphs

Understanding how local update rules amplify small random fluctuations into global consensus is a central challenge in the study of network dynamics and decentralized processes. A prominent family of examples is given by label propagation algorithms, widely used heuristics for community detection whose analysis is complicated by the strong dependencies created as the process evolves. Despite the simplicity and practical success of this class of algorithms, a rigorous understanding of their convergence remains limited. In this paper, we study a specific variant, which we refer to as LPA, on the binomial random graph $\mathcal{G}(n,p)$ - a canonical null model with no planted community structure. We show that LPA reaches consensus asymptotically almost surely when $np\ge n^{5/8+\varepsilon}$ for any fixed $\varepsilon>0$, thus breaking the current best $n^{3/4+\varepsilon}$ analytical barrier. More remarkably, we uncover a phase transition governing \emph{which} label survives: when $np\gg n^{2/3}$, the surviving label is asymptotically almost surely the smallest initial label, whereas for $n^{5/8+\varepsilon}\le np\ll n^{2/3}$, the surviving label is asymptotically almost surely not the smallest one. We further show that, in the critical window $np=Θ(n^{2/3})$, both outcomes have probability $Θ(1).$ Our analysis has several important ingredients. We analyze a staged exploration that progressively reveals information about the random graph while preserving some randomness needed to track the amplification of label advantages across multiple rounds. This analysis is complemented by novel coupling and decoupling arguments that control the dependencies inherent to the process. Along the way, we establish a monotonicity result for competing binomial random variables that we believe is of independent interest.

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arXiv2026

Universality in random graphs via optimal linking systems: trees and beyond

We develop a framework for proving universality results in sparse random graphs. As a first application, we show that there exists an absolute constant $C>1$ such that, with high probability, for every fixed constant $Δ$, the binomial random graph $G(n,C\ln n/n)$ contains every $n$-vertex tree with maximum degree at most $Δ$. This answers a question of Montgomery (Advances in Mathematics, 2019). We also determine, for every $p$ satisfying $C\ln n/n\leq p=n^{-1+o(1)}$, the minimum girth $\ell$ (up to an absolute multiplicative constant) for which with high probability $G(n,p)$ contains all cycle factors of girth at least $Ω(\ell)$. In particular, with high probability $G(n,C\ln n/n)$ contains all cycle factors of girth at least $C\ln n/\ln\ln n$, which is optimal up to a constant factor. This extends an earlier result of Ferber, Kronenberg, and Luh (Transactions of the American Mathematical Society, 2019) and significantly improves a corollary of a deep result of Kahn, Lubetzky, and Wormald (Communications on Pure and Applied Mathematics, 2017). One of the key ingredients in the proofs is establishing the optimal depth of linking systems in sparse random graphs.

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arXiv2026

A hitting-time version of Kotzig's conjecture in random graphs

A family of perfect matchings on the same vertex set is said to have the Kotzig property if the union of any pair of matchings forms a Hamilton cycle. Kotzig famously conjectured that, when $n$ is even, the complete graph on $n$ vertices can be decomposed into $n-1$ perfect matchings with the Kotzig property. A closely related conjecture of Wormald states that, for fixed $d \ge 3$, whp a random $d$-regular graph contains $d$ perfect matchings with the Kotzig property. Recently, the first and third authors considered a version of these problems for the binomial random graph $G(n,p)$ and proved that, for every fixed $k$, there is $C=C(k)$ such that whp $G(n,C\log n/n)$ contains $k$ perfect matchings with the Kotzig property. We strengthen this result by showing a hitting time version: for every fixed $k$, whp the hitting time in the random graph process for finding $k$ edge-disjoint perfect matchings with the Kotzig property is the same as the hitting time for a graph with minimum degree $k$. This is a generalisation of the celebrated result of Bollobás and Frieze for even $n$ establishing the hitting time for $k$ edge-disjoint Hamilton cycles. The proof uses several ideas. We first design a coupling, via a randomised algorithm, of the random graph at the hitting time with $k$ almost-independent perfect matchings and a set of random reservoir edges, which may be of independent interest. We then prove various statistical properties of these matchings, in particular concerning the cycle structure of the $2$-factors formed by pairs of matchings. This includes a novel concept of ``shuffledness'', which we establish through a careful exposure argument and a concentration inequality for random permutations. Finally, we show that the initial $2$-factors can be merged into Hamilton cycles, using the random reservoir edges, by analysing certain Pósa-type rotations via branching processes.

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arXiv2026

Anchored isoperimetry in percolation

We show that, for any infinite transitive graph $G$, the infinite cluster after supercritical percolation almost surely inherits the anchored isoperimetry of $G$. Moreover, the probability of violating the corresponding isoperimetric inequality decays exponentially in the isoperimetric profile of the host graph. This was previously known only for nonamenable transitive graphs, $\mathbb{Z}^d$, and very recently for transitive graphs of polynomial growth. Our proof combines the thinning-and-sprinkling technique with a novel counting algorithm.

math.PR↗
arXiv2026

Cycle lengths in the percolated hypercube

Let $Q^d_p$ be the random subgraph of the $d$-dimensional binary hypercube obtained after edge-percolation with probability $p$. It was shown recently by the authors that, for every $\varepsilon > 0$, there is some $c = c(\varepsilon)>0$ such that, if $pd\ge c$, then typically $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. We strengthen this result to show that, under the same assumptions, typically $Q^d_p$ contains cycles of all even lengths between $4$ and $(1-\varepsilon)2^d$.

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arXiv2026

On the depth of depth-weighted trees

The depth-weighted tree DWT($f$) with weight function $f:\{0,1,2,\ldots\}\to (0,\infty)$ is a dynamic random tree grown from a root $r$ where vertices arrive consecutively and every new vertex attaches to a parent $u$ with probability proportional to $f$(distance between $u$ and $r$). This work is dedicated to a systematic analysis of the depth of DWT($f$). Namely, we provide precise analytic expressions of the typical depth of DWT($f$) for convergent, periodic, slowly growing, and (super-)exponentially growing weight functions. Furthermore, for bounded or exponentially growing $f$, we determine the typical depth up to a multiplicative constant, thus confirming and strengthening a conjecture of Leckey, Mitsche and Wormald.

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arXiv2026

Long induced cycles in pseudorandom graphs

We show that, for some absolute constants $c_1,c_2>0$, every $(n,d,λ)$-graph with $λ\le c_1 d$ contains an induced cycle of length at least $c_2n\log(d/λ)/d$. This is best possible up to the values of $c_1,c_2$. Our techniques include a multi-scale algorithmic analysis, an adapted depth-first exploration procedure, a link to percolation theory, and estimates for random row-and-column extraction in symmetric matrices.

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arXiv2026

Diameter and mixing time of the giant component in the percolated hypercube

We consider bond percolation on the $d$-dimensional binary hypercube with $p=c/d$ for fixed $c>1$. We prove that the typical diameter of the giant component $L_1$ is of order $Θ(d)$, and the typical mixing time of the lazy random walk on $L_1$ is of order $Θ(d^2)$. This resolves long-standing open problems of Bollobás, Kohayakawa and Łuczak from 1994, and of Benjamini and Mossel from 2003. A key component in our approach is a new tight large deviation estimate on the number of vertices in $L_1$ whose proof includes several novel ingredients: a structural description of the residue outside the giant component after sprinkling, a tight quantitative estimate on the spread of the giant in the hypercube, and a stability principle which rules out the disintegration of large connected sets under thinning. This toolkit further allows us to obtain optimal bounds on the expansion in $L_1$.

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arXiv2026

The Mihail-Vazirani conjecture and strong edge-expansion in random $0/1$ polytopes

We study the edge-expansion of the graph of a random $0/1$ polytope $P^d_p$, defined as the convex hull of a random subset of the points in $\{0,1\}^d$ where every point is retained independently and with probability $p$. This problem was introduced more than twenty years ago in a work of Gillmann and Kaibel, and has been extensively studied ever since. We prove that, for every fixed $\varepsilon>0$ and every $p\in(0,1-\varepsilon]$, with high probability the graph of $P^d_p$ has edge-expansion $Θ(d)$. This improves the previously best known bound due to Ferber, Krivelevich, Sales and Samotij, and verifies, in a strong form, the celebrated Mihail-Vazirani conjecture for random $0/1$ polytopes. Although the expansion factor $Θ(d)$ is typically best possible for $p\ge 1/2+\varepsilon$, we also show that the behaviour changes drastically at $p=1/2$. Namely, for every fixed $\varepsilon>0$ and every integer $k\ge 2$, if $p\le 1/2-\varepsilon$, then with high probability the graph of $P^d_p$ has edge-expansion $Ω(d^k)$. Thus, random $0/1$ polytopes exhibit an interesting phase transition at $p=1/2$.

math.CO↗
arXiv2026

Explosive appearance of cores and bootstrap percolation on lattices

Consider the process where the $n$ vertices of a square $2$-dimensional torus appear consecutively in a random order. We show that typically the size of the $3$-core of the corresponding induced unit-distance graph transitions from $0$ to $n-o(n)$ within a single step. Equivalently, by infecting the vertices of the torus in a random order, under two-neighbour bootstrap percolation, the size of the infected set transitions instantaneously from $o(n)$ to $n$. This hitting time result answers a question of Benjamini. We also study the much more challenging and general setting of bootstrap percolation on two-dimensional lattices for a variety of finite-range infection rules. In this case, powerful but fragile bootstrap percolation tools such as the rectangles process and the Aizenman-Lebowitz lemma become unavailable. We develop a new method complementing and replacing these standard techniques, thus allowing us to prove the above hitting time result for a wide family of threshold bootstrap percolation rules on the $2$-dimensional square lattice, including neighbourhoods given by large $\ell^p$ balls for $p\in[1,\infty]$.

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arXiv2025

Sharp thresholds, hitting times and the power of choice for random geometric graphs

We consider a random geometric graph process where random points $(X_i)_{i \ge 1}$ are embedded consecutively in the $d$-dimensional unit torus $\mathbb{T}^d$, and every two points at distance at most $r$ form an edge. As $r\to 0$, we confirm that well-known hitting time results for $k$-connectivity (with $k\ge 1$ fixed) and Hamiltonicity in the Erdős-Rényi graph process also hold for the considered geometric analogue. Moreover, we exhibit a sort of probabilistic monotonicity for each of these properties. We also study a geometric analogue of the power of choice where, at each step, an agent is given two random points sampled independently and uniformly from $\mathbb{T}^d$ and must add exactly one of them to the already constructed point set. When the agent is allowed to make their choice with the knowledge of the entire sequence of random points (offline 2-choice), we show that they can construct a connected graph at the first time $t$ when none of the first $t$ pairs of proposed points contains two isolated vertices in the graph induced by $(X_i)_{i=1}^{2t}$, and maintain connectivity thereafter by following a simple algorithm. We also derive analogous results for $k$-connectivity and Hamiltonicity. This shows that each of the said properties can be attained two times faster (time-wise) and with four times fewer points in the offline 2-choice process compared to the 1-choice process. In the online version where the agent only knows the process until the current time step, we show that $k$-connectivity and Hamiltonicity cannot be significantly accelerated (time-wise) but may be realised on two times fewer points compared to the 1-choice analogue.

math.CO↗
arXiv2025

On the first and second largest components in the percolated Random Geometric Graph

The percolated random geometric graph $G_n(λ, p)$ has vertex set given by a Poisson Point Process in the square $[0,\sqrt{n}]^2$, and every pair of vertices at distance at most 1 independently forms an edge with probability $p$. For a fixed $p$, Penrose proved that there is a critical intensity $λ_c = λ_c(p)$ for the existence of a giant component in $G_n(λ, p)$. Our main result shows that for $λ> λ_c$, the size of the second-largest component is a.a.s. of order $(\log n)^2$. Moreover, we prove that the size of the largest component rescaled by $n$ converges almost surely to a constant, thereby strengthening results of Penrose. We complement our study by showing a certain duality result between percolation thresholds associated to the Poisson intensity and the bond percolation of $G(λ, p)$ (which is the infinite volume version of $G_n(λ,p)$). Moreover, we prove that for a large class of graphs converging in a suitable sense to $G(λ, 1)$, the corresponding critical percolation thresholds converge as well to the ones of $G(λ,1)$.

math.PR↗
arXiv2025

Approximate Itai-Zehavi conjecture for random graphs

A famous conjecture by Itai and Zehavi states that, for every $d$-vertex-connected graph $G$ and every vertex $r$ in $G$, there are $d$ spanning trees of $G$ such that, for every vertex $v$ in $G\setminus \{r\}$, the paths between $r$ and $v$ in different trees are internally vertex-disjoint. We show that with high probability the Itai-Zehavi conjecture holds asymptotically for the Erdős-Rényi random graph $G(n,p)$ when $np= ω(\log n)$ and for random regular graphs $G(n,d)$ when $d= ω(\log n)$. Moreover, we essentially confirm the conjecture up to a constant factor for sparser random regular graphs. This answers positively a question of Draganić and Krivelevich. Our proof makes use of recent developments on sprinkling techniques in random regular graphs.

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arXiv2025

Monotonicity and decompositions of random regular graphs

In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameters $d_1 \leq d_2$, there exists a coupling of $G(n,d_1)$ and $G(n,d_2)$ satisfying that $G(n,d_1) \subseteq G(n,d_2)$ with high probability, confirming a conjecture of Gao, Isaev and McKay in a new regime. Our contributions include new tools for analysing contiguity and total variation distance between random regular graph models, a novel procedure for generating unions of random edge-disjoint perfect matchings, and refined estimates of Gao's bounds on the number of perfect matchings in random regular graphs. In addition, we make progress towards another conjecture of Isaev, McKay, Southwell and Zhukovskii.

math.CO↗
arXiv2025

Spanning trees of bounded degree in random geometric graphs

We determine the sharp threshold for the containment of all $n$-vertex trees of bounded degree in random geometric graphs with $n$ vertices. This provides a geometric counterpart of Montgomery's threshold result for binomial random graphs, and confirms a conjecture of Espuny Díaz, Lichev, Mitsche, and Wesolek. Our proof is algorithmic and adapts to other families of graphs, in particular graphs with bounded genus or tree-width.

math.CO↗
arXiv2025

Nearly spanning cycle in the percolated hypercube

Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant $C=C(\varepsilon)>0$ such that, if $p\ge C/d$, then with high probability $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. This confirms a long-standing folklore conjecture, stated in particular by Condon, Espuny Díaz, Girão, Kühn, and Osthus [Hamiltonicity of random subgraphs of the hypercube, Mem. Amer. Math. Soc. 305 (2024), No. 1534].

math.CO↗
arXiv2025

Universality of the matching number in percolated regular graphs

Fix a sequence of $d$-regular graphs $(G_d)_{d\in \mathbb{N}}$ and denote by $G_{d,p}$ the graph obtained from $G_d$ after edge-percolation with probability $p=c/d$, for a constant $c>0$. We prove a quantitative local convergence of $(G_{d,p})_{d\in \mathbb{N}}$. In combination with results of Bordenave, Lelarge and Salez, it implies that the rescaled matching number of $G_{d,p}$ is asymptotically equivalent to that of the binomial random graph $G(n,c/n)$.

math.CO↗
arXiv2025

On the $(k+2,k)$-problem of Brown, Erdős and Sós for $k=5,6,7$

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no subgraph with $k$ edges and at most $s$ vertices. Brown, Erdős and Sós [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press 1973] conjectured that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. The value of the limit was previously determined for $k=2$ in the original paper of Brown, Erdős and Sós, for $k=3$ by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for $k=4$ by Glock, Joos, Kim, Kühn, Lichev and Pikhurko [Proc. Amer. Math. Soc., Series B, 11 (2024) 173-186] while Delcourt and Postle [Proc. Amer. Math. Soc., 152 (2024), 1881-1891] proved the conjecture (without determining the limiting value). In this paper, we determine the value of the limit in the Brown-Erdős-Sós Problem for $k\in \{5,6,7\}$. More generally, we obtain the value of $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all $r\geq 3$ and $k\in \{5,6,7\}$. In addition, by combining these new values with recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new asymptotic values for several generalised Ramsey numbers.

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