arXiv subjects

Lyuben Lichev

Publications and source records attributed to Lyuben Lichev.

50 records · Page 3Linked to original sources

arXiv2023

On the $(6,4)$-problem of Brown, Erdős and Sós

Let $f^{(r)}(n;s,k)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$ vertices not containing a subgraph with $k$ edges and at most $s$ vertices. In 1973, Brown, Erdős and Sós conjectured that the limit $$\lim_{n\to \infty} n^{-2} f^{(3)}(n;k+2,k)$$ exists for all $k$ and confirmed it for $k=2$. Recently, Glock showed this for $k=3$. We settle the next open case, $k=4$, by showing that $f^{(3)}(n;6,4)=\left(\frac{7}{36}+o(1)\right)n^2$ as $n\to\infty$. More generally, for all $k\in \{3,4\}$, $r\ge 3$ and $t\in [2,r-1]$, we compute the value of the limit $\lim_{n\to \infty} n^{-t}f^{(r)}(n;k(r-t)+t,k)$, which settles a problem of Shangguan and Tamo.

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arXiv2023

Color-avoiding percolation of random graphs: between the subcritical and the intermediate regime

Fix a graph $G$ in which every edge is colored in some of $k\ge 2$ colors. Two vertices $u$ and $v$ are CA-connected if $u$ and $v$ may be connected using any subset of $k - 1$ colors. CA-connectivity is an equivalence relation dividing the vertex set into classes called CA-components. In two recent papers, Ráth, Varga, Fekete, and Molontay, and Lichev and Schapira studied the size of the largest CA-component in a randomly colored random graph. The second of these works distinguished and studied three regimes (supercritical, intermediate, and subcritical) in which the largest CA-component has respectively linear, logarithmic, and bounded size. In this short note, we describe the phase transition between the intermediate and the subcritical regime.

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arXiv2022

Lower bound on the running time of Pop-Stack Sorting on a random permutation

Pop-Stack Sorting is an algorithm that takes a permutation as an input and sorts its elements. It consists of several steps. At one step, the algorithm reads the permutation it has to process from left to right and reverses each of its maximal decreasing subsequences of consecutive elements. It terminates at the first step that outputs the identity permutation. In this note, we answer a question of Defant on the running time of Pop-Stack Sorting on the uniform random permutation $σ_n$. More precisely, we show that there is a constant $c > 0.5$ such that asymptotically almost surely, the algorithm needs at least $cn$ steps to terminate on $σ_n$.

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arXiv2022

Localization Game for Random Geometric Graphs

The localization game is a two player combinatorial game played on a graph $G=(V,E)$. The cops choose a set of vertices $S_1 \subseteq V$ with $|S_1|=k$. The robber then chooses a vertex $v \in V$ whose location is hidden from the cops, but the cops learn the graph distance between the current position of the robber and the vertices in $S_1$. If this information is sufficient to locate the robber, the cops win immediately; otherwise the cops choose another set of vertices $S_2 \subseteq V$ with $|S_2|=k$, and the robber may move to a neighbouring vertex. The new distances are presented to the robber, and if the cops can deduce the new location of the robber based on all information they accumulated thus far, then they win; otherwise, a new round begins. If the robber has a strategy to avoid being captured, then she wins. The localization number is defined to be the smallest integer $k$ so that the cops win the game. In this paper we determine the localization number (up to poly-logarithmic factors) of the random geometric graph $G \in \mathcal G(n,r)$ slightly above the connectivity threshold.

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arXiv2022

Conflict-free hypergraph matchings

A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost-regular, uniform hypergraph $\mathcal{H}$ with small maximum codegree has an almost-perfect matching. We extend this result by obtaining a ``conflict-free'' matching, where conflicts are encoded via a collection $\mathcal{C}$ of subsets $C\subseteq E(\mathcal{H})$. We say that a matching $\mathcal{M}\subseteq E(\mathcal{H})$ is conflict-free if $\mathcal{M}$ does not contain an element of $\mathcal{C}$ as a subset. Under natural assumptions on $\mathcal{C}$, we prove that $\mathcal{H}$ has a conflict-free, almost-perfect matching. This has many applications, one of which yields new asymptotic results for so-called ``high-girth'' Steiner systems. Our main tool is a random greedy algorithm which we call the ``conflict-free matching process''.

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arXiv2021

Local boxicity

A box is the cartesian product of real intervals, which are either bounded or equal to $\mathbb{R}$. A box is said to be $d$-local if at most $d$ of the intervals are bounded. In this paper, we investigate the recently introduced local boxicity of a graph $G$, which is the minimum $d$ such that $G$ can be represented as the intersection of $d$-local boxes in some dimension. We prove that all graphs of maximum degree $Δ$ have local boxicity $O(Δ)$, while almost all graphs of maximum degree $Δ$ have local boxicity $Ω(Δ)$, improving known upper and lower bounds. We also give improved bounds on the local boxicity as a function of the number of edges or the genus. Finally, we investigate local boxicity through the lens of chromatic graph theory. We prove that the family of graphs of local boxicity at most 2 is $χ$-bounded, which means that the chromatic number of the graphs in this class can be bounded by a function of their clique number. This extends a classical result on graphs of boxicity at most 2.

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arXiv2021

The giant component after percolation of product graphs

In this paper we show the existence of a sharp threshold for the appearance of a giant component after percolation of Cartesian products of graphs under assumptions on their maximum degrees and their isoperimetric constants. In particular, this generalises a work of Ajtai, Komlós and Szemerédi from 1982 concerning percolation of the hypercube in high dimension.

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arXiv2021

On the power of choice for Boolean functions

In this paper we consider a variant of the well-known Achlioptas process for graphs adapted to monotone Boolean functions. Fix a number of choices $r\in \mathbb N$ and a sequence of increasing functions $(f_n)_{n\ge 1}$ such that, for every $n\ge 1$, $f_n:\{0,1\}^n\mapsto \{0,1\}$. Given $n$ bits which are all initially equal to 0, at each step $r$ 0-bits are sampled uniformly at random and are proposed to an agent. Then, the agent selects one of the proposed bits and turns it from 0 to 1 with the goal to reach the preimage of 1 as quickly as possible. We nearly characterize the conditions under which an acceleration by a factor of $r(1+o(1))$ is possible, and underline the wide applicability of our results by giving examples from the fields of Boolean functions and graph theory.

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arXiv2021

The jump of the clique chromatic number of random graphs

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Pralat noted that around p \approx n^{-1/2} the clique chromatic number of the random graph G_{n,p} changes by n^{Ω(1)} when we increase the edge-probability p by n^{o(1)}, but left the details of this surprising phenomenon as an open problem. We settle this problem, i.e., resolve the nature of this polynomial `jump' of the clique chromatic number of the random graph G_{n,p} around edge-probability p \approx n^{-1/2}. Our proof uses a mix of approximation and concentration arguments, which enables us to (i) go beyond Janson's inequality used in previous work and (ii) determine the clique chromatic number of G_{n,p} up to logarithmic factors for any edge-probability p.

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arXiv2021

On the chromatic number of the preferential attachment graph

We prove that for every $m\in \mathbb N$ and every $δ\in (-m,0)$, the chromatic number of the preferential attachment graph $PA_t(m, δ)$ is asymptotically almost surely equal to $m+1$. The proof relies on a combinatorial construction of a family of digraphs of chromatic number $m+1$ followed by a proof that asymptotically almost surely there is a digraph in this family, which is realised as a subgraph of the preferential attachment graph.

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arXiv2021

On the Boxicity of Kneser Graphs and Complements of Line Graphs

An axis-parallel $d$-dimensional box is a cartesian product $I_1\times I_2\times \dots \times I_b$ where $I_i$ is a closed sub-interval of the real line. For a graph $G = (V,E)$, the $boxicity \ of \ G$, denoted by $\text{box}(G)$, is the minimum dimension $d$ such that $G$ is the intersection graph of a family $(B_v)_{v\in V}$ of $d$-dimensional boxes in $\mathbb R^d$. Let $k$ and $n$ be two positive integers such that $n\geq 2k+1$. The $Kneser \ graph$ $Kn(k,n)$ is the graph with vertex set given by all subsets of $\{1,2,\dots,n\}$ of size $k$ where two vertices are adjacent if their corresponding $k$-sets are disjoint. In this note, we derive a general upper bound for $\text{box}(Kn(k,n))$, and a lower bound in the case $n\ge 2k^3-2k^2+1$, which matches the upper bound up to an additive factor of $Θ(k^2)$. Our second contribution is to provide upper and lower bounds for the boxicity of the complement of the line graph of any graph $G$, and as a corollary, we derive that $\text{box}(Kn(2,n))\in \{n-3, n-2\}$ for every $n\ge 5$.

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arXiv2021

A note on the Erdős-Szekeres theorem in two dimensions

Burkill and Mirsky, and Kalmanson, prove independently that, for every $r\ge 2, n\ge 1$, there is a sequence of $r^{2^n}$ vectors in $\mathbb R^n$, which does not contain a subsequence of $r+1$ vectors $v^1, v^2,\dots,v^{r+1}$ such that, for every $i$ between 1 and $n$, $(v^{j}_i)_{1\le j\le r+1}$ forms a monotone sequence. Moreover, $r^{2^n}$ is the largest integer with this property. In this short note, for two vectors $u = (u_1, u_2,\dots, u_n)$ and $v = (v_1, v_2, \dots, v_n)$ in $\mathbb R^n$, we say that $u\le v$ if, for every $i$ between 1 and $n$, $u_i\le v_i$. Just like Burkill and Mirsky, and Kalmanson, for every $k, \ell\ge 1, d\ge 2$ we find the maximal $N_1, N_2$ (which turn out to be equal) such that there are numerical two-dimensional arrays of size $(k+\ell-1)\times N_1$ and $(k+\ell)\times N_2$, which neither contain a subarray of size $k\times d$, whose columns form a non-decreasing sequence of $d$ vectors in $\mathbb R^k$, nor contain a subarray of size $\ell\times d$, whose columns form a non-increasing sequence of $d$ vectors in $\mathbb R^{\ell}$. In a consequent discussion, we consider a generalisation of this setting and make a connection with a famous problem in coding theory.

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arXiv2020

On the Decycling Number of $4$-regular Random Graphs

The decycling number $ϕ(G)$ of a graph $G$ is the smallest number of vertices which can be removed from $G$ so that the resulting graph has no cycles. Bau, Wormald and Zhou conjectured that with probability tending to one the decycling number of the random $4$-regular graph $G_4(n)$ on $n$ vertices is equal to $\lceil (n+1)/3\rceil$. In this paper we show that this conjecture holds asymptotically, i.e. asymptotically almost surely $\lim_{n \to \infty}ϕ(G_4(n))/n = 1/3$.

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arXiv2020

New Constructions related to the Polynomial Sphere Recognition Problem

We construct a simply connected $2-$complex $C$ embeddable in $3-$space such that for any embedding of $C$ in $\mathbb S^3$, any edge contraction forms a minor of the $2-$complex not embeddable in $3-$space. We achieve this by proving that every edge of $C$ forms a nontrivial knot in any of the embeddings of $C$ in $\mathbb S^3$.

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