arXiv subjects

Balázs Keszegh

Publications and source records attributed to Balázs Keszegh.

At least 19 recordsLinked to original sources

arXiv2026

Saturation and semisaturation of oriented graphs

We consider a natural extension of the classical saturation problem to oriented graphs, defined as the minimum number of edges in an oriented graph $H$ on $n$ vertices that avoids a fixed oriented graph $G$, but adding any missing edge to $H$ creates a copy of $G$. Our main result shows that saturation functions of oriented graphs attain at least four distinct orders of magnitude. In particular, if $G$ is acyclic, then its saturation function is bounded if and only if $G$ contains an isolated edge; otherwise, it is linear. Certain tournaments together with an isolated edge give further infinite examples of bounded and linear saturation functions. Two additional orders of magnitude are $Θ(n \sqrt{\log n})$ and $Θ(n \log n)$. For these two, the lower bounds are obtained by adapting a recently developed approach for the saturation problem in edge-ordered graphs, while the matching upper bounds are constructive. Finally, we study the semisaturation problem, in which we require only that adding any missing edge to $H$ creates a copy of $G$. We completely characterize the possible orders of magnitude: the semisaturation function is either bounded, linear, or $Θ(n \log n)$.

math.CO↗

arXiv2026

Saturation of edge-ordered graphs

For an edge-ordered graph $G$, an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with an arbitrary label to $H$ creates a copy of $G$. The saturation function is the minimum number of edges in a $G$-saturated graph. For (unordered) graphs, $0$-$1$ matrices, and vertex-ordered graphs, the saturation function is always $O(n)$ and satisfies a dichotomy: it is either $O(1)$ or $Θ(n)$. The saturation function of an edge-ordered graph follows a weaker dichotomy, being either $O(1)$ or $Ω(n)$. However, by finding edge-ordered graphs whose saturation functions are $Ω(n \sqrt{\log n})$, we show that $O(n)$ is not a universal upper bound. We also study the semisaturation problem for edge-ordered graphs, a variant of the saturation problem in which $H$ is not required to be $G$-free. We prove a general upper bound $O(n \log n)$ and characterize edge-ordered graphs with bounded semisaturation functions. We then present several families of edge-ordered graphs with bounded, linear, and superlinear (semi)saturation functions. We also introduce a natural variant of saturation in which the added edge is required to receive the smallest label. The behaviour of the two variants is similar in many respects, which motivated us to investigate the second variant extensively.

math.CO↗

arXiv2026

Forbidden stars in multidimensional $0$-$1$ matrices and visibility of lattice points

A $d$-dimensional $0$-$1$ matrix $M$ of size $n_1\times n_2\times \dots \times n_d$ can be considered as a Boolean function $M: B(n_1\times n_2\times \dots \times n_d) \to \{ 0,1\}$, where $B$ is the $d$-dimensional box of lattice points $(x_1, \dots, , x_d)\in Z^d$ with $0\leq x_i \leq n_i-1$, $1\leq i\leq d$. The $0$-$1$ matrix $M$ can also be described as a subset $P:=P(M)$ of $B$ such that $x\in P$ if and only if $M(x)=1$. A $k$-star with center $p$ in $M$ corresponds to a $(k+1)$-element subset $\{ p, p_1, \dots , p_k\} \subset B$ such that $p$ and $p_i$ differ only in one coordinate (for all $1\leq i\leq k$) and these $k$ coordinates are distinct. Here we consider the problem of determining the maximum number of $1$-entries of a $0$-$1$ matrix $M$ of dimension $d$ and size $n\times n \times \dots \times n$ that avoids all $k$-stars. Our main results are the asymptotical solution of the problem for every $d$ and $k$ (as $n\to \infty$), very close bounds for $k=d$, and the exact solution of the $d=k=3$ case. This problem has connections to several other areas of discrete mathematics, including $k$-partite hypergraphs, independent set problems, dominating set problems and covering codes. One of our tools (concerning maximal packings of induced copies of a given hypergraph) might have independent interest.

math.CO↗

arXiv2026

Piercing all maximum cliques in hypergraphs

Graphs whose maximum clique size exceeds half of the total number of vertices satisfy a classical property: the family of their maximum sized cliques can be pierced by a single vertex. This result dates back to a 1965 theorem by Hajnal. Motivated by this theorem, Jung, Keszegh, Pálvölgyi, and Yuditsky recently conjectured that an analogous result should hold for hypergraphs of larger uniformity, with an appropriate constant replacing the threshold $1/2$. In this paper we refute this conjecture in a strong form. We show that for any constant $c<1$ and integers $k\ge 3$ and $t\ge 1$, there exist $k$-uniform hypergraphs $G$ whose maximum clique size exceeds $c|V(G)|$, yet the family of maximum size cliques of $G$ cannot be pierced by $t$ vertices. This demonstrates that no universal constant threshold guarantees bounded piercing number for maximum cliques in uniform hypergraphs. We discuss further questions concerning the relationship between clique size and piercing maximum cliques in hypergraphs, and introduce a geometric variant of the problem using Helly's Theorem.

math.CO↗

arXiv2026

On the number of tangencies among $1$-intersecting $x$-monotone curves

Let $\cC$ be a set of curves in the plane such that no three curves in $\cC$ intersect at a single point and every pair of curves in $\cC$ intersect at exactly one point which is either a crossing or a touching point. János Pach conjectured that the number of pairs of curves in $\cC$ that touch each other is $O(|\cC|)$. We prove this conjecture for $x$-monotone curves.

math.CO↗

arXiv2026

The Zarankiewicz Problem for Polygon Visibility Graphs

We prove a quasi-linear upper bound on the size of $K_{t,t}$-free polygon visibility graphs. For visibility graphs of star-shaped and monotone polygons we show a linear bound. In the more general setting of $n$ points on a simple closed curve and visibility pseudo-segments, we provide an $O(n \log n)$ upper bound and an $Ω(nα(n))$ lower bound.

cs.CG↗

arXiv2026

On the maximum number of tangencies among $1$-intersecting curves

According to a conjecture of Pach, there are $O(n)$ tangent pairs among any family of $n$ Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two special cases, however, for the general case the currently best upper bound is only $O(n^{7/4})$. This is also the best known bound on the number of tangencies in the relaxed case where every pair of arcs has \emph{at most} one common point. We improve the bounds for the latter and former cases to $O(n^{5/3})$ and $O(n^{3/2})$, respectively. We also consider a few other variants of these questions, for example, we show that if the arcs are \emph{$x$-monotone}, each pair intersects at most once and their left endpoints lie on a common vertical line, then the maximum number of tangencies is $Θ(n^{4/3})$. Without this last condition the number of tangencies is $O(n^{4/3}(\log n)^{1/3})$, improving a previous bound of Pach and Sharir. Along the way we prove a graph-theoretic theorem which extends a result of Erdős and Simonovits and may be of independent interest.

math.CO↗

arXiv2026

Unavoidable patterns and plane paths in dense topological graphs

Let $C_{s,t}$ be the complete bipartite geometric graph, with $s$ and $t$ vertices on two distinct parallel lines respectively, and all $s t$ straight-line edges drawn between them. In this paper, we show that every complete bipartite simple topological graph, with parts of size $2(k-1)^4 + 1$ and $2^{k^{5k}}$, contains a topological subgraph weakly isomorphic to $C_{k,k}$. As a corollary, every $n$-vertex simple topological graph not containing a plane path of length $k$ has at most $O_k(n^{2 - 8/k^4})$ edges. When $k = 3$, we obtain a stronger bound by showing that every $n$-vertex simple topological graph not containing a plane path of length 3 has at most $O(n^{4/3})$ edges. We also prove that $x$-monotone simple topological graphs not containing a plane path of length 3 have at most a linear number of edges.

math.CO↗

arXiv2026

On Triangles in Colored Pseudoline Arrangements

We consider the faces in pseudoline arrangements in which the pseudolines are colored with two colors. Björner, Las Vergnas, Sturmfels, White, and Ziegler conjecture the existence of a two-colored triangle in such arrangements. We consider variants of this problem. We show that in any non-trivial two-coloring of a pseudoline arrangement there exists a two-colored triangle or quadrangle. We also investigate the existence of a bichromatic triangle assuming certain structures on the coloring. Previously, several authors investigated the chromatic number and independence number of hypergraphs whose vertices correspond to the pseudolines of an arrangement and the hyperedges correspond to the faces of the arrangement. We show that the maximum of the independence numbers of such hypergraphs is $\lceil \frac{2}{3}n-1\rceil$. We also prove that if we only consider the triangular faces then this maximum becomes $n-Θ(\log n)$.

math.CO↗

arXiv2025

Coloring Geometric Hypergraphs: A Survey

The \emph{chromatic number} of a hypergraph is the smallest number of colors needed to color the vertices such that no edge of at least two vertices is monochromatic. Given a family of geometric objects $\mathcal{F}$ that covers a subset $S$ of the Euclidean space, we can associate it with a hypergraph whose vertex set is $\mathcal F$ and whose edges are those subsets ${\mathcal{F}'}\subset \mathcal F$ for which there exists a point $p\in S$ such that ${\mathcal F}'$ consists of precisely those elements of $\mathcal{F}$ that contain $p$. The question whether $\mathcal F$ can be split into 2 coverings is equivalent to asking whether the chromatic number of the hypergraph is equal to 2. There are a number of competing notions of the chromatic number that lead to deep combinatorial questions already for abstract hypergraphs. In this paper, we concentrate on \emph{geometrically defined} (in short, \emph{geometric}) hypergraphs, and survey many recent coloring results related to them. In particular, we study and survey the following problem, dual to the above covering question. Given a set of points $S$ in the Euclidean space and a family $\mathcal{F}$ of geometric objects of a fixed type, define a hypergraph ${\mathcal H}_m$ on the point set $S$, whose edges are the subsets of $S$ that can be obtained as the intersection of $S$ with a member of $\mathcal F$ and have at least $m$ elements. Is it true that if $m$ is large enough, then the chromatic number of ${\mathcal H}_m$ is equal to 2?

math.CO↗

arXiv2025

Orientation of good covers

We study systems of orientations on triples that satisfy the following so-called interiority condition: $\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ implies $\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O; we denote a P3O realizable by points as p-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-T3O, is the order type of a point set in general position. In our paper "Orientation of convex sets" we defined a 3-order on pairwise intersecting convex sets; such a P3O is called a C-P3O. In this paper we extend this 3-order to pairwise intersecting good covers; such a P3O is called a GC-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a C-T3O and a GC-T3O, respectively. The main result of this paper is that there is a p-T3O that is not a GC-T3O, implying also that it is not a C-T3O -- this latter problem was left open in our earlier paper. Our proof involves several combinatorial and geometric observations that can be of independent interest. Along the way, we define several further special families of GC-T3O's.

math.CO↗

arXiv2025

The complexity of recognizing $ABAB$-free hypergraphs

The study of geometric hypergraphs gave rise to the notion of $ABAB$-free hypergraphs. A hypergraph $\mathcal{H}$ is called $ABAB$-free if there is an ordering of its vertices such that there are no hyperedges $A,B$ and vertices $v_1,v_2,v_3,v_4$ in this order satisfying $v_1,v_3\in A\setminus B$ and $v_2,v_4\in B\setminus A$. In this paper, we prove that it is NP-complete to decide if a hypergraph is $ABAB$-free. We show a number of analogous results for hypergraphs with similar forbidden patterns, such as $ABABA$-free hypergraphs. As an application, we show that deciding whether a hypergraph is realizable as the incidence hypergraph of points and pseudodisks is also NP-complete.

math.CO↗

arXiv2025

On the number of edges in saturated partial embeddings of maximal planar graphs

We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane subgraph of $G$ such that the addition of any edge either violates planarity or results in a graph that is not a subgraph of $G$. We focus on maximal planar graphs and establish an upper bound on $\mathrm{sat}_{\mathcal{P}}(G)$ by showing there exists a universal constant $ε> 0$ such that $\mathrm{sat}_{\mathcal{P}}(G) < (3-ε)v(G)$ for any maximal planar graph $G$ with $v(G) \geq 16$. This answers a question posed by Clifton and Simon. Additionally, we derive lower bound results and demonstrate that for maximal planar graphs with sufficiently large number of vertices, the minimum ratio $\mathrm{sat}_{\mathcal{P}}(G)/e(G)$ lies within the interval $(1/16, 1/9 + o(1)]$.

math.CO↗

arXiv2024

The maximum number of digons formed by pairwise crossing pseudocircles

In 1972, Branko Grünbaum conjectured that any arrangement of $n>2$ pairwise crossing pseudocircles in the plane can have at most $2n-2$ digons (regions enclosed by exactly two pseudoarcs), with the bound being tight. While this conjecture has been confirmed for cylindrical arrangements of pseudocircles and more recently for geometric circles, we extend these results to any simple arrangement of pairwise intersecting pseudocircles. Using techniques from the above-mentioned special cases, we provide a complete proof of Grünbaum's conjecture that has stood open for over five decades.

math.CO↗

arXiv2024

On dual-ABAB-free and related hypergraphs

Geometric motivations warranted the study of hypergraphs on ordered vertices that have no pair of hyperedges that induce an alternation of some given length. Such hypergraphs are called ABA-free, ABAB-free and so on. Since then various coloring and other combinatorial results were proved about these families of hypergraphs. We prove a characterization in terms of their incidence matrices which avoids using the ordering of the vertices. Using this characterization, we prove new results about the dual hypergraphs of ABAB-free hypergraphs. In particular, we show that dual-ABAB-free hypergraphs are not always proper $2$-colorable even if we restrict ourselves to hyperedges that are larger than some parameter $m$.

math.CO↗

arXiv2024

Query complexity of Boolean functions on the middle slice of the cube

We study the query complexity on slices of Boolean functions. Among other results we show that there exists a Boolean function for which we need to query all but 7 input bits to compute its value, even if we know beforehand that the number of 0's and 1's in the input are the same, i.e., when our input is from the middle slice. This answers a question of Byramji. Our proof is non-constructive, but we also propose a concrete candidate function that might have the above property. Our results are related to certain natural discrepancy type questions that, somewhat surprisingly, have not been studied before.

math.CO↗

arXiv2024

On the number of digons in arrangements of pairwise intersecting circles

A long-standing open conjecture of Branko Grünbaum from 1972 states that any simple arrangement of $n$ pairwise intersecting pseudocircles in the plane can have at most $2n-2$ digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Grünbaum's conjecture is true for arrangements of pairwise intersecting pseudocircles in which there are three pseudocircles every pair of which create a digon. In this paper we prove this over 50-year-old conjecture of Grünbaum for any simple arrangement of pairwise intersecting circles in the plane.

math.CO↗

arXiv2024

Orientation of convex sets

We introduce a novel definition of orientation on the triples of a family of pairwise intersecting planar convex sets and study its properties. In particular, we compare it to other systems of orientations on triples that satisfy a so-called interiority condition: $\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ imply $\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O; we denote a P3O realizable by points as p-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-T3O, is the order type of a point set in general position. Despite these similarities to order types, P3O and p-T3O that can arise from the orientation of pairwise intersecting convex sets, denoted by C-P3O and C-T3O, turn out to be quite different from order types: there is no containment relation among the family of all C-P3O's and the family of all p-P3O's, or among the families of C-T3O's and p-T3O's. Finally, we study properties of these orientations if we also require that the family of underlying convex sets satisfies the (4,3) property.

math.CO↗