arXiv subjects

Balázs Patkós

Publications and source records attributed to Balázs Patkós.

84 records · Page 5Linked to original sources

arXiv2016

The minimum number of vertices in uniform hypergraphs with given domination number

The \textit{domination number} $γ(\mathcal{H})$ of a hypergraph $\mathcal{H}=(V(\mathcal{H}),E(\mathcal{H})$ is the minimum size of a subset $D\subset V(\mathcal{H}$ of the vertices such that for every $v\in V(\mathcal{H})\setminus D$ there exist a vertex $d \in D$ and an edge $H\in E(\mathcal{H})$ with $v,d\in H$. We address the problem of finding the minimum number $n(k,γ)$ of vertices that a $k$-uniform hypergraph $\mathcal{H}$ can have if $γ(\mathcal{H})\ge γ$ and $\mathcal{H}$ does not contain isolated vertices. We prove that $$n(k,γ)=k+Θ(k^{1-1/γ})$$ and also consider the $s$-wise dominating and the distance-$l$ dominating version of the problem. In particular, we show that the minimum number $n_{dc}(k,γ, l)$ of vertices that a connected $k$-uniform hypergraph with distance-$l$ domination number $γ$ can have is roughly $\frac{kγl}{2}$

math.CO↗

arXiv2016

Dominating sequences in grid-like and toroidal graphs

A 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 $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.

math.CO↗

arXiv2015

On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method

We study the function $M(n,k)$ which denotes the number of maximal $k$-uniform intersecting families $F\subseteq \binom{[n]}{k}$. Improving a bound of Balogh at al. on $M(n,k)$, we determine the order of magnitude of $\log M(n,k)$ by proving that for any fixed $k$, $M(n,k) =n^{Θ(\binom{2k}{k})}$ holds. Our proof is based on Tuza's set pair approach. The main idea is to bound the size of the largest possible point set of a cross-intersecting system. We also introduce and investigate some related functions and parameters.

math.CO↗

arXiv2015

Avoider-Enforcer star games

In 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. We analyse both versions of Avoider-Enforcer games -- the strict and the monotone -- and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases $f^{mon}_\mathcal{F}$, $f^-_\mathcal{F}$ and $f^+_\mathcal{F}$, where $\mathcal{F}$ is the hypergraph of the game.

math.CO↗

arXiv2014

Search Problems in Vector Spaces

We 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 subspace of $V$. We will be interested in determining the minimum number of queries that is needed to find $\mathbf{v}$ provided all queries are subspaces of $V$ and the answer to a query $U$ is YES if $\mathbf{v} \leqslant U$ and NO if $\mathbf{v} \not\leqslant U$. This number will be denoted by $A(n,q)$ in the adaptive case (when for each queries answers are obtained immediately and later queries might depend on previous answers) and $M(n,q)$ in the non-adaptive case (when all queries must be made in advance). In the case $n=3$ we prove $2q-1=A(3,q)<M(3,q)$ if $q$ is large enough. While for general values of $n$ and $q$ we establish the bounds \[ n\log q \le A(n,q) \le (1+o(1))nq \] and \[ (1-o(1))nq \le M(n,q) \le 2nq, \] provided $q$ tends to infinity.

math.CO↗

arXiv2013

Nonrepetitive colorings of lexicographic product of graphs

A 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 by substituting every vertex of $G$ by a copy of $H$, and every edge of $G$ by a copy of $K_{k,k}$. %Our main results are the following. We prove that for a sufficiently long path $P$, a nonrepetitive coloring of $P[K_k]$ needs at least $3k+\lfloor k/2\rfloor$ colors. If $k>2$ then we need exactly $2k+1$ colors to nonrepetitively color $P[E_k]$, where $E_k$ is the empty graph on $k$ vertices. If we further require that every copy of $E_k$ be rainbow-colored and the path $P$ is sufficiently long, then the smallest number of colors needed for $P[E_k]$ is at least $3k+1$ and at most $3k+\lceil k/2\rceil$. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.

math.CO↗

arXiv2012

Majority and Plurality Problems

Given 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. We address the problem of finding the minimum number of queries (a comparison of a pair of balls if they have the same color or not) that is needed to decide whether a majority, k-majority or plurality ball exists and if so then show one such ball. We consider both adaptive and non-adaptive strategies and in certain cases, we also address weighted versions of the problems.

math.CO↗

arXiv2011

Two-part set systems

The 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 $A \cap X_i=B \cap X_i$ for some $i$, then $|\cF| \le {n \choose \lfloor n/2 \rfloor}$. We consider variations of this problem by replacing the Sperner property with the intersection property and considering families that satisfiy various combinations of these properties on one or both parts $X_1$, $X_2$. Along the way, we prove the following new result which may be of independent interest: let $\cF, \cG$ be families of subsets of an $n$-element set such that $\cF$ and $\cG$ are both intersecting and cross-Sperner, meaning that if $A \in \cF$ and $B \in \cG$, then $A \not\subset B$ and $B \not\subset A$. Then $|\cF| +|\cG| < 2^{n-1}$ and there are exponentially many examples showing that this bound is tight.

math.CO↗

arXiv2011

On the ratio of maximum and minimum degree in maximal intersecting families

To study how balanced or unbalanced a maximal intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is we consider the ratio $\mathcal{R}(\mathcal{F})=\frac{Δ(\mathcal{F})}{δ(\mathcal{F})}$ of its maximum and minimum degree. We determine the order of magnitude of the function $m(n,r)$, the minimum possible value of $\mathcal{R}(\mathcal{F})$, and establish some lower and upper bounds on the function $M(n,r)$, the maximum possible value of $\mathcal{R}(\mathcal{F})$. To obtain constructions that show the bounds on $m(n,r)$ we use a theorem of Blokhuis on the minimum size of a non-trivial blocking set in projective planes.

math.CO↗

arXiv2011

Saturating Sperner families

A 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 size of a family saturating the $k$-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of $l$-sets and $(l+1)$-sets.

math.CO↗

arXiv2011

Cross-Sperner families

A 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 $|\cF|+|\cG|$ or with the product $|\cF|\cdot |\cG|$. We show that if $\cF, \cG \subseteq 2^{[n]}$, then $|\cF||\cG| \le 2^{2n-4}$ and $|\cF|+|\cG|$ is maximal if $\cF$ or $\cG$ consists of exactly one set of size $\lceil n/2 \rceil$ provided the size of the ground set $n$ is large enough and both $\cF$ and $\cG$ are non-empty.

math.CO↗

arXiv2010

Large B_d-free and union-free subfamilies

For a property $Γ$ and a family of sets $\cF$, let $f(\cF,Γ)$ be the size of the largest subfamily of $\cF$ having property $Γ$. For a positive integer $m$, let $f(m,Γ)$ be the minimum of $f(\cF,Γ)$ over all families of size $m$. A family $\cF$ is said to be $B_d$-free if it has no subfamily $\cF'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinct sets such that for every $I,J \subseteq [d]$, both $F_I \cup F_J=F_{I \cup J}$ and $F_I \cap F_J = F_{I \cap J}$ hold. A family $\cF$ is $a$-union free if $F_1\cup ... F_a \neq F_{a+1}$ whenever $F_1,..,F_{a+1}$ are distinct sets in $\FF$. We verify a conjecture of Erd\H os and Shelah that $f(m, B_2\text{\rm -free})=Θ(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\text{\rm -free})$ and $f(m,a\text{\rm -union free})$.

math.CO↗