Results for "Balázs Patkós"
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On the general position problem on Kneser graphsMar 19 2019In a graph $G$, a geodesic between two vertices $x$ and $y$ is a shortest path connecting $x$ to $y$. A subset $S$ of the vertices of $G$ is in general position if no vertex of $S$ lies on any geodesic between two other vertices of $S$. The size of a ... More Distribution of colors in Gallai coloringsMar 11 2019A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $n_1\ge n_2 \ge \dots \ge n_k$ with $\sum_{i=1}^kn_k={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $n_i$ ... More Nonrepetitive colorings of lexicographic product of graphsOct 20 2012Sep 16 2013A coloring $c$ of the vertices of a graph $G$ is nonrepetitive if there exists no path $v_1v_2\ldots v_{2l}$ for which $c(v_i)=c(v_{l+i})$ for all $1\le i\le l$. Given graphs $G$ and $H$ with $|V(H)|=k$, the lexicographic product $G[H]$ is the graph obtained ... More Effective Models of the Electroweak Phase TransitionJun 06 1995The consecutive integration over the distinct mass scales ${\cal O}(T),{\cal O}(gT)$ leads to a hierarchy of effective models for the electroweak phase transition. Different techniques for the realisation of such strategy are reviewed. Advantages and ... More Search Problems in Vector SpacesSep 26 2013Jan 30 2014We consider the following $q$-analog of the basic combinatorial search problem: let $q$ be a prime power and $\GF(q)$ the finite field of $q$ elements. Let $V$ denote an $n$-dimensional vector space over $\GF(q)$ and let $\mathbf{v}$ be an unknown 1-dimensional ... More Distribution of colors in Gallai coloringsMar 11 2019Mar 16 2019A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1\ge e_2 \ge \dots \ge e_k$ with $\sum_{i=1}^ke_i={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ ... More Finding non-minority balls with majority and plurality queriesDec 20 2018Given a set of $n$ colored balls, a \textit{majority, non-minority or plurality ball} is one whose color class has size more than $n/2$, at least $n/2$ or larger than any other color class, respectively. We describe linear time algorithms for finding ... More Line Percolation in Finite Projective PlanesAug 01 2016We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional results on the ... More On general position sets in Cartesian gridsJul 10 2019Jul 25 2019The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three pairwise distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. A formula ... More t-wise Berge and t-heavy hypergraphsFeb 08 2019In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph $\mathcal{H}$ ... More Distribution of colors in Gallai coloringsMar 11 2019Apr 16 2019A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1\ge e_2 \ge \dots \ge e_k$ with $\sum_{i=1}^ke_i={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ ... More Distribution of colors in Gallai coloringsMar 11 2019Jul 26 2019A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1\ge e_2 \ge \dots \ge e_k$ with $\sum_{i=1}^ke_i={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ ... More Cross-Sperner familiesApr 20 2011A pair of families $(\cF,\cG)$ is said to be \emph{cross-Sperner} if there exists no pair of sets $F \in \cF, G \in \cG$ with $F \subseteq G$ or $G \subseteq F$. There are two ways to measure the size of the pair $(\cF,\cG)$: with the sum $|\cF|+|\cG|$ ... More Majority and Plurality ProblemsMar 07 2012Given a set of n balls each colored with a color, a ball is said to be majority, k-majority, plurality if its color class has size larger than half of the number of balls, has size at least k, has size larger than any other color class; respectively. ... More Large B_d-free and union-free subfamiliesDec 17 2010For a property $\Gamma$ and a family of sets $\cF$, let $f(\cF,\Gamma)$ be the size of the largest subfamily of $\cF$ having property $\Gamma$. For a positive integer $m$, let $f(m,\Gamma)$ be the minimum of $f(\cF,\Gamma)$ over all families of size $m$. ... More Stability results on vertex Turán problems in Kneser graphsApr 11 2018Apr 20 2018The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free, was recently ... More Stability results on vertex Turán problems in Kneser graphsApr 11 2018Mar 07 2019The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free, was recently ... More Domination game on uniform hypergraphsOct 01 2017In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph $\mathcal{H}$ by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the set of vertices ... More Finding a non-minority ball with majority answersSep 28 2015Sep 28 2016Suppose we are given a set of $n$ balls $\{b_1,\ldots,b_n\}$ each colored either red or blue in some way unknown to us. To find out some information about the colors, we can query any triple of balls $\{b_{i_1},b_{i_2},b_{i_3}\}$. As an answer to such ... More Saturating Sperner familiesMay 23 2011A family $\cF \subseteq 2^{[n]}$ saturates the monotone decreasing property $\cP$ if $\cF$ satisfies $\cP$ and one cannot add any set to $\cF$ such that property $\cP$ is still satisfied by the resulting family. We address the problem of finding the minimum ... More Rainbow Ramsey problems for the Boolean latticeSep 23 2018We address the following rainbow Ramsey problem: For posets $P,Q$ what is the smallest number $n$ such that any coloring of the elements of the Boolean lattice $B_n$ either admits a monochromatic copy of $P$ or a rainbow copy of $Q$. We consider both ... More Avoider-Enforcer star gamesFeb 11 2013Jan 30 2015In this paper, we study $(1 : b)$ Avoider-Enforcer games played on the edge set of the complete graph on $n$ vertices. For every constant $k\geq 3$ we analyse the $k$-star game, where Avoider tries to avoid claiming $k$ edges incident to the same vertex. ... More Grundy dominating sequences and zero forcing setsFeb 02 2017In a graph $G$ a sequence $v_1,v_2,\dots,v_m$ of vertices is Grundy dominating if for all $2\le i \le m$ we have $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ and is Grundy total dominating if for all $2\le i \le m$ we have $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. ... More Dominating sequences in grid-like and toroidal graphsJul 01 2016A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of ... More Vertex Turán problems for the oriented hypercubeJul 18 2018In this short note we consider the oriented vertex Tur\'an problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the vertices of the oriented ... More Two-part set systemsOct 01 2011The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\cF$ is a family of subsets of $X$ such that no two sets $A, B \in \cF$ satisfy $A \subset B$ (or $B \subset A$) and $A \cap ... More Forbidding rank-preserving copies of a posetOct 25 2017The maximum size, $La(n,P)$, of a family of subsets of $[n]=\{1,2,...,n\}$ without containing a copy of $P$ as a subposet, has been intensively studied. Let $P$ be a graded poset. We say that a family $\mathcal{F}$ of subsets of $[n]=\{1,2,...,n\}$ contains ... More On the number of containments in $P$-free familiesApr 04 2018A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq \mathcal F$ is a copy of the poset $P$ if there exists a bijection $i:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$. A family $\mathcal F$ is $P$-free, if it does ... More Computing arbitrary Lagrangian Eulerian maps for evolving surfacesDec 06 2016Mar 29 2017The good mesh quality of an evolving discretized surface or domain is often compromised during time evolution. In recent years this phenomena have been overcome in a couple of ways, one of them uses arbitrary Lagrangian Eulerian maps. However, the numerical ... More Accumulation Points of Graphs of Baire-1 and Baire-2 FunctionsDec 01 2015Jun 11 2017During the last few decades E. S. Thomas, S. J. Agronsky, J. G. Ceder, and T. L. Pearson gave an equivalent definition of the real Baire class 1 functions by characterizing their graph. In this paper, using their results, we consider the following problem: ... More The Wiedemann-Franz law in the SU(N) Wolff modelFeb 16 2006We study the electrical and thermal transport through the SU(N) Wolff model with the use of bosonization. The Wilson ratio reaches unity as N grows to infinity. The electric conductance is dominated by the charge channel, and decreases monotonically with ... More Covering Paths and Trees for Planar GridsNov 03 2013Given a set of points in the plane, a covering path is a polygonal path that visits all the points. In this paper we consider covering paths of the vertices of an n x m grid. We show that the minimal number of segments of such a path is $2\min(n,m)-1$ ... More Boundary effect on CDW: Friedel oscillations, STM imageMar 17 2005We study the effect of open boundary condition on charge density waves (CDW). The electron density oscillates rapidly close to the boundary, and additional non-oscillating terms (~ln(r)) appear. The Friedel oscillations survive beyond the CDW coherence ... More Spin supplementary conditions for spinning compact binariesSep 06 2016We consider the different spin supplementary conditions (SSC) for a spinning compact binary with the leading-order spin-orbit (SO) interaction. The Lagrangian of the binary system can be constructed but it is acceleration-dependent in two cases of SSC. ... More Algebraic degrees of pseudo-Anosov stretch factorsJun 21 2015Nov 19 2016The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface $S$ can be as high as the dimension of the Teichm\"uller space of $S$. In addition to proving this, we completely ... More Characterizations and Properties of Graphs of Baire FunctionsNov 02 2016Let $X$ be a paracompact topological space and $Y$ be a Banach space. In this paper, we will characterize the Baire-1 functions $f:X\rightarrow{Y}$ by their graph: namely, we will show that $f$ is a Baire-1 function if and only if its graph $gr(f)$ is ... More Baire categorical aspects of first passage percolationNov 21 2017In the previous decades, the theory of first passage percolation became a highly important area of probability theory. In this work, we will observe what can be said about the corresponding structure if we forget about the probability measure defined ... More Mixing time of an unaligned Gibbs sampler on the squareJun 27 2017Oct 06 2018The paper concerns a particular example of the Gibbs sampler and its mixing efficiency. Coordinates of a point are rerandomized in the unit square $[0,1]^2$ to approach a stationary distribution with density proportional to $\exp(-A^2(u-v)^2)$ for $(u,v)\in ... More On the Volume of Boolean expressions of Large Congruent BallsDec 21 2017We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of $r$, which will ... More On the Ledrappier-Young formula for self-affine measuresMar 03 2015Jun 02 2015Ledrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier-Young ... More An implementation of the relational k-means algorithmApr 25 2013A C# implementation of a generalized k-means variant called relational k-means is described here. Relational k-means is a generalization of the well-known k-means clustering method which works for non-Euclidean scenarios as well. The input is an arbitrary ... More The Fermi edge singularity in the SU(N) Wolff modelSep 15 2005The low temperature properties of the SU(N) Wolff impurity model are studied via Abelian bosonization. The path integral treatment of the problem allows for an exact evaluation of low temperature properties of the model. The single particle Green's function ... More Baire categorical aspects of first passage percolation IINov 11 2018In this paper we continue our earlier work about topological first passage percolation and answer certain questions asked in our previous paper. Notably, we prove that apart from trivialities, in the generic configuration there exists exactly one geodesic ... More Algebraic degrees of pseudo-Anosov stretch factorsJun 21 2015Oct 16 2018The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface $S$ can be as high as the dimension of the Teichm\"uller space of $S$. In addition to proving this, we completely ... More Analysis of a non-reversible Markov chain speedup by a single edgeMay 08 2019Jun 06 2019We present a Markov chain example where non-reversibility and an added edge jointly improve mixing time: when a random edge is added to a cycle of $n$ vertices and a Markov chain with a drift is introduced, we get mixing time of $O(n^{3/2})$ with probability ... More The variety of domination gamesJul 07 2018Domination game [SIAM J.\ Discrete Math.\ 24 (2010) 979--991] and total domination game [Graphs Combin.\ 31 (2015) 1453--1462] are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination ... More On Grundy total domination number in product graphsDec 23 2017A longest sequence $(v_1,\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \setminus \bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ of the sequence is called the Grundy total domination ... More Magnetotransport and thermoelectricity in disordered grapheneJan 29 2007We have studied the electric and thermal response of two-dimensional Dirac-fermions in a quantizing magnetic field in the presence of localized disorder. The electric and heat current operators in the presence of magnetic field are derived. The self-energy ... More NMR relaxation rate and static spin susceptibility in grapheneFeb 17 2008The NMR relaxation rate and the static spin susceptibility in graphene are studied within a tight-binding description. At half filling, the NMR relaxation rate follows a power law as $T^2$ on the particle-hole symmetric side, while with a finite chemical ... More Improved mixing rates of directed cycles by added connectionSep 04 2015Sep 16 2015We provide bounds on the mixing rate of a Markov chain whose mixing properties are improved by a combination of adding long distance edges and introducing non-reversibility. Our model is built on a cycle graph and involves selecting a sublinear number ... More Multidimensional Sampling of Isotropically Bandlimited SignalsMar 01 2017A new lower bound on the average reconstruction error variance of multidimensional sampling and reconstruction is presented. It applies to sampling on arbitrary lattices in arbitrary dimensions, assuming a stochastic process with constant, isotropically ... More On the threshold of spread-out voter model percolationMay 17 2017Oct 02 2017In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z}^d$ has state (or 'opinion') 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from ... More Planar S-systems: PermanenceMay 25 2018We characterize permanence of planar S-systems. Further, we construct a planar S-system with three limit cycles. A measure-theoretic approach to the theory of dense hypergraphsOct 22 2008Oct 27 2008In this paper we develop a measure-theoretic method to treat problems in hypergraph theory. Our central theorem is a correspondence principle between three objects: An increasing hypergraph sequence, a measurable set in an ultraproduct space and a measurable ... More Testing of sequences by simulationNov 30 2010Let $\xi$ be a random integer vector, having uniform distribution \[\mathbf{P} \{\xi = (i_1,i_2,...,i_n) = 1/n^n \} \ \hbox{for} \ 1 \leq i_1,i_2,...,i_n\leq n.\] A realization $(i_1,i_2,...,i_n)$ of $\xi$ is called \textit{good}, if its elements are ... More Harmonic Manifolds and the Volume of Tubes about CurvesJun 08 2015Oct 03 2016H. Hotelling proved that in the n-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric ... More Reentrant Kondo effect in Landau quantized grapheneMay 14 2007We have studied the interplay of an Anderson impurity in Landau quantized graphene, with special emphasis on the influence of the chemical potential. Within the slave-boson mean-field theory, we found reentrant Kondo behaviour by varying the chemical ... More