arXiv subjects

Zoltán Lóránt Nagy

Publications and source records attributed to Zoltán Lóránt Nagy.

At least 37 records · Page 2Linked to original sources

arXiv2024

Partitioning the projective plane to two incidence-rich parts

An internal or friendly partition of a vertex set $V(G)$ of a graph $G$ is a partition to two nonempty sets $A\cup B$ such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbor set in its own class and the neighbor set of the other class, that can be attained for all vertices at the same time for the incidence graphs of desarguesian planes of square order.

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arXiv2024

Multicolor Turán numbers II -- a generalization of the Ruzsa-Szemerédi theorem and new results on cliques and odd cycles

In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem. Let $ex_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. The case when both $F$ and $G$ are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when $F$ and $G$ are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear $r$-uniform hypergraph Turán problems to determine $ex_r^{lin}(n,G)$ form a class of the multicolor Turán problem, following the identity $ex_r^{lin}(n,G)=ex_{K_r}(n,G)$, our results determine the linear hypergraph Turán numbers of every graph of girth $3$ and for every $r$ up to a subpolynomial factor. Furthermore, when $G$ is a triangle, we settle the case $F=C_5$ and give bounds for the cases $F=C_{2k+1}$, $k\ge 3$ as well.

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arXiv2022

The extensible No-Three-In-Line problem

The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek to select sets of large density from the infinite grid $Z^{2}$ while avoiding a collinear triple. We show the existence of such a set which contains $Θ(n/\log^{1+\varepsilon}n)$ points in $[1,n]^{2}$ for all $n$, where $\varepsilon>0$ is an arbitrarily small real number. We also give computational evidence suggesting that a set of lattice points may exist that has at least $n/2$ points on every large enough $n\times n$ grid.

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arXiv2021

Multicolor Turán numbers

We consider a natural generalisation of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem by studying the maximum number $ex_F(n,G)$ of edge-disjoint copies of a fixed graph $F$ can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. We determine the pairs $\{F, G\}$ for which the order of magnitude of $ex_F(n,G)$ is quadratic and prove several asymptotic results using various tools from the regularity lemma and supersaturation to graph packing results.

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arXiv2021

Unified approach to the generalized Turán problem and supersaturation

In this paper we introduce a unifying approach to the generalized Turán problem and supersaturation results in graph theory. The supersaturation-extremal function $satex(n, F : m, G)$ is the least number of copies of a subgraph $G$ an $n$-vertex graph can have, which contains at least $m$ copies of $F$ as a subgraph. We present a survey, discuss previously known results and obtain several new ones focusing mainly on proof methods, extremal structure and phase transition phenomena. Finally we point out some relation with extremal questions concerning hypergraphs, particularly Berge-type results.

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arXiv2021

A note on internal partitions: the $5$-regular case and beyond

An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the $5$-regular case and show that among the subgraphs of minimum degree at least $3$, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all $5$-regular Abelian Cayley graphs which do not have an internal partition.

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arXiv2021

On the Turán number of the blow-up of the hexagon

The $r$-blowup of a graph $F$, denoted by $F[r]$, is the graph obtained by replacing the vertices and edges of $F$ with independent sets of size $r$ and copies of $K_{r,r}$, respectively. For bipartite graphs $F$, very little is known about the order of magnitude of the Turán number of $F[r]$. In this paper we prove that $\mathrm{ex}(n,C_6[2])=O(n^{5/3})$ and, more generally, for any positive integer $t$, $\mathrm{ex}(n,θ_{3,t}[2])=O(n^{5/3})$. This is tight when $t$ is sufficiently large.

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arXiv2021

Short minimal codes and covering codes via strong blocking sets in projective spaces

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension $k$ over the $q$-element Galois field which is linear in both $q$ and $k$, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well. The contributions rely on geometric and probabilistic arguments.

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arXiv2021

Steiner triple systems and spreading sets in projective spaces

We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper subsystem which contains it. We give sharp upper bounds on the size of a minimal spreading set in a Steiner triple system and show that if all the minimal spreading sets are large then the examined triple system must be a projective space. We also show that the size of a minimal spreading set is not an invariant of a Steiner triple system.

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arXiv2021

Generalized Outerplanar Turán numbers and maximum number of k-vertex subtrees

We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees of given order. This problem turns out to be equivalent to determine the maximum number of k+2-cycles in n-vertex outerplanar graphs, thus we settle the generalized outerplanar Turán number for all cycles. We also determine the exponential growth of the generalized outerplanar Turán number of paths Pk as a function of k which implies the order of magnitude of the generalized outerplanar Turán number of arbitrary trees. The bounds are strongly related to the sequence of Catalan numbers.

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arXiv2020

Coloring linear hypergraphs: the Erdős-Faber-Lovász conjecture and the Combinatorial Nullstellensatz

The long-standing Erdős-Faber-Lovász conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős-Faber-Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations, which verifies a necessary condition for our algebraic approach to work.

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arXiv2020

On the balanced upper chromatic number of finite projective planes

In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph $H$, the maximum number $k$ for which there is a balanced rainbow-free $k$-coloring of $H$ is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane $\mathrm{PG}(2,q)$ for all $q$. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.

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arXiv2020

Triangle areas in line arrangements

A widely investigated subject in combinatorial geometry, originated from Erdős, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between $Ω(n^2)$ and $O(n^{9/4})$. This result is strongly connected to both additive combinatorial results and Szemerédi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.

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arXiv2020

Spreading linear triple systems and expander triple systems

The existence of Steiner triple systems STS(n) of order n containing no nontrivial subsystem is well known for every admissible n. We generalize this result in two ways. First we define the expander property of 3-uniform hypergraphs and show the existence of Steiner triple systems which are almost perfect expanders. Next we define the strong and weak spreading property of linear hypergraphs, and determine the minimum size of a linear triple system with these properties, up to a small constant factor. This property is strongly connected to the connectivity of the structure and of the so-called influence maximization. We also discuss how the results are related to Erdős' conjecture on locally sparse STSs, influence maximization, subsquare-free Latin squares and possible applications in finite geometry.

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arXiv2019

The Turán number of blow-ups of trees

A conjecture of Erdős from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this bound holds for a large family of $r$-degenerate bipartite graphs, including all $r$-degenerate blow-ups of trees. Our results generalise many previously proven cases of the Erdős conjecture, including the related results of Füredi and Alon, Krivelevich and Sudakov. Our proof uses supersaturation and a random walk on an auxiliary graph.

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arXiv2017

Saturating sets in projective planes and hypergraph covers

Let $Π_q$ be an arbitrary finite projective plane of order $q$. A subset $S$ of its points is called saturating if any point outside $S$ is collinear with a pair of points from $S$. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to $\lceil\sqrt{3q\ln{q}}\rceil+ \lceil(\sqrt{q}+1)/2\rceil$. The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.

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arXiv2017

Supersaturation of $C_4$: from Zarankiewicz towards Erdős-Simonovits-Sidorenko

For a positive integer $n$, a graph $F$ and a bipartite graph $G\subseteq K_{n,n}$ let ${F(n+n, G)}$ denote the number of copies of $F$ in $G$, and let $F(n+n, m)$ denote the minimum number of copies of $F$ in all graphs $G\subseteq K_{n,n}$ with $m$ edges. The study of such a function is the subject of theorems of supersaturated graphs and closely related to the Sidorenko-Erdős-Simonovits conjecture as well. In the present paper we investigate the case when $F= K_{2,t}$ and in particular the quadrilateral graph case. For $F=C_4$, we obtain exact results if $m$ and the corresponding Zarankiewicz number differ by at most $n$, by a finite geometric construction of almost difference sets. $F= K_{2,t}$ if $m$ and the corresponding Zarankiewicz number differs by $Cn\sqrt{n}$ we prove asymptotically sharp results. We also study stability questions and point out the connections to covering and packing block designs.

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arXiv2017

Coupon-Coloring and total domination in Hamiltonian planar triangulations

We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning.

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