arXiv subjects

Lyuben Lichev

Publications and source records attributed to Lyuben Lichev.

At least 37 records · Page 2Linked to original sources

arXiv2024

Color-avoiding percolation on the Erdős-Rényi random graph

We consider a recently introduced model of color-avoiding percolation defined as follows. Every edge in a graph $G$ is colored in some of $k\ge 2$ colors. Two vertices $u$ and $v$ in $G$ are said to be CA-connected if $u$ and $v$ may be connected using any subset of $k-1$ colors. CA-connectivity defines an equivalence relation on the vertex set of $G$ whose classes are called CA-components. We study the component structure of a randomly colored Erdős-Rényi random graph of constant average degree. We distinguish three regimes for the size of the largest component: a supercritical regime, a so-called intermediate regime, and a subcritical regime, in which the largest CA-component has respectively linear, logarithmic, and bounded size. Interestingly, in the subcritical regime, the bound is deterministic and given by the number of colors.

math.PR↗

arXiv2024

Discrepancies of spanning trees in dense graphs

We address several related problems on combinatorial discrepancy of trees in a setting introduced by Erdős, Füredi, Loebl and Sós. Given a fixed tree $T$ on $n$ vertices and an edge-colouring of the complete graph $K_n$, for every colour, we find a copy of $T$ in $K_n$ where the number of edges in that colour significantly exceeds its expected count in a uniformly random embedding. This resolves a problem posed by Erdős, Füredi, Loebl and Sós by generalising their work from two to many colours. Furthermore, if $T$ has maximum degree $Δ\leqεn$ for sufficiently small $ε> 0$ and the edge-colouring of $K_n$ is both balanced and ``not too close'' to one particular instance, we show that, for every colour, there is a copy of $T$ in $K_n$ where that colour appears on linearly more edges than any other colour. Several related examples are provided to demonstrate the necessity of the introduced structural restrictions. Our proofs combine saturation arguments for the existence of particular coloured substructures and analysis of conveniently defined local exchanges. Using similar methods, we investigate the existence of copies of a graph $H$ with prescribed number of edges in each colour in $2$-edge-coloured dense host graphs. In particular, for a graph $H$ with bounded maximum degree and balanced $2$-edge-colourings $\mathbf{c}$ of a host graph $G$ with minimum degree at least $(1-ε)n$ for some $ε> 0$, we show that, for any sufficiently large $n$ and sufficiently small $ε$, there exists a copy of $H$ where the number of edges in the two colours differ by at most $2$. Moreover, we completely characterise the pairs $(H,\mathbf{c})$ for which the difference of $2$ cannot be improved, refuting a conjecture by Mohr, Pardey, and Rautenbach.

math.CO↗

arXiv2024

Dynamic random graphs with vertex removal

We introduce and analyse a Dynamic Random Graph with Vertex Removal (DRGVR) defined as follows. At every step, with probability $p > 1/2$ a new vertex is introduced, and with probability $1-p$ a vertex, chosen uniformly at random among the present ones (if any), is removed from the graph together with all edges adjacent to it. In the former case, the new vertex connects by an edge to every other vertex with probability inversely proportional to the number of vertices already present. We prove that the DRGVR converges to a local limit and determine this limit. Moreover, we analyse its component structure and distinguish a subcritical and a supercritical regime with respect to the existence of a giant component. As a byproduct of this analysis, we obtain upper and lower bounds for the critical parameter. Furthermore, we provide precise expression of the maximum degree (as well as in- and out-degree for a natural orientation of the DRGVR). Several concentration and stability results complete the study.

math.PR↗

arXiv2024

$d$-connectivity of the random graph with restricted budget

In this short note, we consider a graph process recently introduced by Frieze, Krivelevich and Michaeli. In their model, the edges of the complete graph $K_n$ are ordered uniformly at random and are then revealed consecutively to a player called Builder. At every round, Builder must decide if they accept the edge proposed at this round or not. We prove that, for every $d\ge 2$, Builder can construct a spanning $d$-connected graph after $(1+o(1))n\log n/2$ rounds by accepting $(1+o(1))dn/2$ edges with probability converging to 1 as $n\to \infty$. This settles a conjecture of Frieze, Krivelevich and Michaeli.

math.CO↗

arXiv2024

Vertex-separating path systems in random graphs

A set $V$ is said to be separated by subsets $V_1,\ldots,V_k$ if, for every pair of distinct elements of $V$, there is a set $V_i$ that contains exactly one of them. Imposing structural constraints on the separating subsets is often necessary for practical purposes and leads to a number of fascinating (and, in some cases, already classical) graph-theoretic problems. In this work, we are interested in separating the vertices of a random graph by path-connected vertex sets $V_1,\ldots,V_k$, jointly forming a separating system. First, we determine the size of the smallest separating system of $G(n,p)$ when $np\to \infty$ up to lower order terms, and exhibit a threshold phenomenon around the sharp threshold for connectivity. Second, we show that random regular graphs of sufficiently high degree can typically be optimally separated by $\lceil \log_2 n\rceil$ sets. Moreover, we provide bounds for the minimum degree threshold for optimal separation of general graphs.

math.CO↗

arXiv2024

On the local resilience of random geometric graphs with respect to connectivity and long cycles

Given an increasing graph property $\mathcal{P}$, a graph $G$ is $α$-resilient with respect to $\mathcal{P}$ if, for every spanning subgraph $H\subseteq G$ where each vertex keeps more than a $(1-α)$-proportion of its neighbours, $H$ has property $\mathcal{P}$. We study the above notion of local resilience with $G$ being a random geometric graph $G_d(n,r)$ obtained by embedding $n$ vertices independently and uniformly at random in $[0,1]^d$, and connecting two vertices by an edge if the distance between them is at most $r$. First, we focus on connectivity. We show that, for every $\varepsilon>0$, for $r$ a constant factor above the sharp threshold for connectivity $r_c$ of $G_d(n,r)$, the random geometric graph is $(1/2-\varepsilon)$-resilient for the property of being $k$-connected, with $k$ of the same order as the expected degree. However, contrary to binomial random graphs, for sufficiently small $\varepsilon>0$, connectivity is not born $(1/2-\varepsilon)$-resilient in $2$-dimensional random geometric graphs. Second, we study local resilience with respect to the property of containing long cycles. We show that, for $r$ a constant factor above $r_c$, $G_d(n,r)$ is $(1/2-\varepsilon)$-resilient with respect to containing cycles of all lengths between constant and $2n/3$. Proving $(1/2-\varepsilon)$-resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that $G_d(n,r)$ is $α$-resilient with respect to Hamiltonicity for a fixed constant $α= α(d)<1/2$.

math.CO↗

arXiv2024

Local dimer dynamics in higher dimensions

We consider local dynamics of the dimer model (perfect matchings) on hypercubic boxes $[n]^d$. These consist of successively switching the dimers along alternating cycles of prescribed (small) lengths. We study the connectivity properties of the dimer configuration space equipped with these transitions. Answering a question of Freire, Klivans, Milet and Saldanha, we show that in three dimensions any configuration admits an alternating cycle of length at most 6. We further establish that any configuration on $[n]^d$ features order $n^{d-2}$ alternating cycles of length at most $4d-2$. We also prove that the dynamics of dimer configurations on the unit hypercube of dimension $d$ is ergodic when switching alternating cycles of length at most $4d-4$. Finally, in the planar but non-bipartite case, we show that parallelogram-shaped boxes in the triangular lattice are ergodic for switching alternating cycles of lengths 4 and 6 only, thus improving a result of Kenyon and Rémila, which also uses 8-cycles. None of our proofs make reference to height functions.

math.CO↗

arXiv2024

The maximal running time of hypergraph bootstrap percolation

We show that for every $r\ge 3$, the maximal running time of the $K^{r}_{r+1}$-bootstrap percolation in the complete $r$-uniform hypergraph on $n$ vertices $K_n^r$ is $Θ(n^r)$. This answers a recent question of Noel and Ranganathan in the affirmative, and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form $r^{-r} \mathrm{e}^{O(r)}$ as $r\to\infty$.

math.CO↗

arXiv2024

Percolation on dense random graphs with given degrees

In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied. Far from being able to classify all such degree sequences, we exhibit several new threshold phenomena for the order of the largest component in terms of both sources of randomness. We also provide an example of a degree sequence for which the order of the largest component undergoes an unbounded number of jumps in terms of the percolation parameter, giving rise to a behavior that cannot be observed without percolation.

math.PR↗

arXiv2024

Zero-sum Random Games on Directed Graphs

This paper considers a class of two-player zero-sum games on directed graphs whose vertices are equipped with random payoffs of bounded support known by both players. Starting from a fixed vertex, players take turns to move a token along the edges of the graph. On the one hand, for acyclic directed graphs of bounded degree and sub-exponential expansion, we show that the value of the game converges almost surely to a constant at an exponential rate dominated in terms of the expansion. On the other hand, for the infinite $d$-ary tree that does not fall into the previous class of graphs, we show convergence at a double-exponential rate in terms of the expansion.

math.OC↗

arXiv2023

Large monochromatic components in colorings of complete hypergraphs

Gyárfás famously showed that in every $r$-coloring of the edges of the complete graph $K_n$, there is a monochromatic connected component with at least $\frac{n}{r-1}$ vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for $k$-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of $1+o(1)$ as the number of colors grows. We further generalize these questions to ask about counts of vertex $s$-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.

math.CO↗

arXiv2023

Annulus graphs in $\mathbb R^d$

A $d$-dimensional annulus graph with radii $R_1$ and $R_2$ (here $R_2\ge R_1\ge 0$) is a graph embeddable in $\mathbb R^d$ so that two vertices $u$ and $v$ form an edge if and only if their images in the embedding are at distance in the interval $[R_1, R_2]$. In this paper we show that the family $\mathcal A_d(R_1,R_2)$ of $d$-dimensional annulus graphs with radii $R_1$ and $R_2$ is uniquely characterised by $R_2/R_1$ when this ratio is sufficiently large. Moreover, as a step towards a better understanding of the structure of $\mathcal A_d(R_1,R_2)$, we show that $\sup_{G\in \mathcal A_d(R_1,R_2)} χ(G)/ω(G)$ is given by $\exp(O(d))$ for all $R_1,R_2$ satisfying $R_2\ge R_1 > 0$ and also $\exp(Ω(d))$ if moreover $R_2/R_1\ge 1.2$.

math.CO↗

arXiv2023

Brownian snails with removal die out in one dimension

Brownian snails with removal is a spatial epidemic model defined as follows. Initially, a homogeneous Poisson process of susceptible particles on $\mathbb R^d$ with intensity $λ>0$ is deposited and a single infected one is added at the origin. Each particle performs an independent standard Brownian motion. Each susceptible particle is infected immediately when it is within distance 1 from an infected particle. Each infected particle is removed at rate $α>0$, and removed particles remain such forever. Answering a question of Grimmett and Li, we prove that in one dimension, for all values of $λ$ and $α$, the infection almost surely dies out.

math.PR↗

arXiv2023

New results for the random nearest neighbor tree

In this paper, we study the online nearest neighbor random tree in dimension $d\in \mathbb N$ (called $d$-NN tree for short) defined as follows. We fix the torus $\mathbb T^d_n$ of dimension $d$ and area $n$ and equip it with the metric inherited from the Euclidean metric in $\mathbb R^d$. Then, embed consecutively $n$ vertices in $\mathbb T^d_n$ uniformly at random and independently, and let each vertex but the first one connect to its (already embedded) nearest neighbor. Call the resulting graph $G_n$. We show multiple results concerning the degree sequence of $G_n$. First, we prove that typically the number of vertices of degree at least $k\in \mathbb N$ in the $d$-NN tree decreases exponentially with $k$ and is tightly concentrated by a new Lipschitz-type concentration inequality that may be of independent interest. Second, we obtain that the maximum degree of $G_n$ is of logarithmic order. Third, we give explicit bounds for the number of leaves that are independent of the dimension and also give estimates for the number of paths of length two. Moreover, we show that typically the height of a uniformly chosen vertex in $G_n$ is $(1+o(1))\log n$ and the diameter of $\mathbb T^d_n$ is $(2e+o(1))\log n$, independently of the dimension. Finally, we define a natural infinite analog $G_{\infty}$ of $G_n$ and show that it corresponds to the local limit of the sequence of finite graphs $(G_n)_{n \ge 1}$. Moreover, we prove almost surely that $G_{\infty}$ is locally finite, that the simple random walk on $G_{\infty}$ is recurrent, and that $G_{\infty}$ is connected.

math.PR↗

arXiv2023

On the modularity of 3-regular random graphs and random graphs with given degree sequences

The modularity of a graph is a parameter that measures its community structure; the higher its value (between $0$ and $1$), the more clustered the graph is. In this paper we show that the modularity of a random $3$-regular graph is at least $0.667026$ asymptotically almost surely (a.a.s.), thereby proving a conjecture of McDiarmid and Skerman. We also improve the a.a.s. upper bound given therein to $0.789998$. For a uniformly chosen graph $G_n$ over a given bounded degree sequence with average degree $d(G_n)$ and with $|CC(G_n)|$ many connected components, we distinguish two regimes with respect to the existence of a giant component. In the subcritical regime, we compute the second term of the modularity. In the supercritical regime, we prove that there is $\varepsilon > 0$, for which the modularity is a.a.s. at least \begin{equation*} \dfrac{2\left(1 - μ\right)}{d(G_n)}+\varepsilon, \end{equation*} where $μ$ is the asymptotically almost sure limit of $\dfrac{|CC(G_n)|}{n}$.

math.PR↗

arXiv2023

The semi-random tree process

The online semi-random graph process is a one-player game which starts with the empty graph on $n$ vertices. At every round, a player (called Builder) is presented with a vertex $v$ chosen uniformly at random and independently from previous rounds, and constructs an edge of their choice that is incident to $v$. Inspired by recent advances on the semi-random graph process, we define a family of generalised online semi-random models. We analyse a particular instance that shares similar features with the original semi-random graph process and determine the hitting times of the classical graph properties minimum degree $k$, $k$-connectivity, containment of a perfect matching, a Hamiltonian cycle and an $H$-factor for a fixed graph $H$ possessing an additional tree-like property. Along the way, we derive a few consequences of the famous Aldous-Broder algorithm that may be of independent interest.

math.CO↗

arXiv2023

On the minimum bisection of random $3$-regular graphs

In this paper we give new bounds on the bisection width of random 3-regular graphs on $n$ vertices. The main contribution is a new lower bound of $0.103295n$ based on a first moment method together with a structural analysis of the graph, thereby improving a 27-year-old result of Kostochka and Melnikov. We also give a complementary upper bound of $0.139822n$ by combining a result of Lyons with original combinatorial insights. Developping this approach further, we obtain a non-rigorous improved upper bound with the help of Monte Carlo simulations.

math.PR↗

arXiv2023

Sharp threshold for embedding balanced spanning trees in random geometric graphs

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph $\mathcal{G}(n,r,d)$. In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.

math.CO↗